A Tight Upper Bound on the Second-Order Coding Rate of the Parallel Gaussian Channel with Feedback
Silas~L.~Fong and Raymond~W.~Yeung
~\IEEEmembership{Fellow
~IEEE
Silas~L.~Fong and Vincent~Y.~F.~Tan% <-this % stops a space \thanks{S.~L.~Fong and V.~Y.~F.~Tan were supported by NUS Young Investigator Award under Grant R-263-000-B37-133. %and in part by Ministry of Education (MOE) Tier 2 Grant R-263-000-B61-112.
....lastname \thanks{...
This paper investigates the asymptotic expansion for the maximum rate of fixed-length codes over a parallel Gaussian channel with feedback under the following setting: A peak power constraint is imposed on every transmitted codeword, and the average error probabilities of decoding the transmitted message are non-vanishing as the blocklength increases. %It is well known that the presence of feedback does not increase the first-order asymptotics of the channel, i.e., capacity, in the asymptotic expansion, and the closed-form expression of the capacity can be obtained by the well-known water-filling algorithm. The main contribution of this paper is a self-contained proof of an upper bound on the first- and second-order asymptotics of the parallel Gaussian channel with feedback. The proof techniques involve developing an information spectrum bound followed by using Curtiss' theorem to show that a sum of dependent random variables associated with the information spectrum bound converges in distribution to a sum of independent random variables, thus facilitating the use of the usual central limit theorem. Combined with existing achievability results, our result implies that the presence of feedback does not improve the first- and second-order asymptotics. %\boldmath
ParallelGaussianFBr3.tex

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%
% paper title
% can use linebreaks \\ within to get better formatting as desired
\title{A Tight Upper Bound on the Second-Order Coding Rate of the Parallel Gaussian Channel with Feedback}
%
%
% author names and IEEE memberships
% note positions of commas and nonbreaking spaces ( ~ ) LaTeX will not break
% a structure at a ~ so this keeps an author's name from being broken across
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%\thanks{Silas~L.~Fong is with the Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong (e-mail: lhfong5@ie.cuhk.edu.hk).}% <-this % stops a space
%\thanks{Raymond~W.~Yeung is with the Institute of Network Coding and the Department
%of Information Engineering, The Chinese University of Hong Kong,
%N.T., Hong Kong (e-mail: whyeung@ie.cuhk.edu.hk).}}% <-this % stops a space
%%\thanks{Manuscript received April 19, 2005; revised January 11, 2007.}}

\author{Silas~L.~Fong and Vincent~Y.~F.~Tan% <-this % stops a space
\thanks{S.~L.~Fong and V.~Y.~F.~Tan were supported by NUS Young Investigator Award under Grant R-263-000-B37-133.
 %and in part by Ministry of Education (MOE) Tier 2 Grant R-263-000-B61-112.}%
 }%
\thanks{S.~L.~Fong is with the Department of Electrical and Computer Engineering, NUS, Singapore 117583 (e-mail: \texttt{silas\_fong@nus.edu.sg}).}%
\thanks{V.~Y.~F.~Tan is with the Department of Electrical and Computer Engineering, NUS, Singapore 117583, and also with the Department of Mathematics, NUS, Singapore 119076 (e-mail: \texttt{vtan@nus.edu.sg}).}}% <-this % stops a space
%\thanks{Manuscript received April 19, 2005; revised January 11, 2007.}}

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\begin{abstract}
This paper investigates the asymptotic expansion for the maximum rate of fixed-length codes over a parallel Gaussian channel with feedback under the following setting: A peak power constraint is imposed on every transmitted codeword, and the average error probabilities of decoding the transmitted message are non-vanishing as the blocklength increases.
%It is well known that the presence of feedback does not increase the first-order asymptotics of the channel, i.e., capacity, in the asymptotic expansion, and the closed-form expression of the capacity can be obtained by the well-known water-filling algorithm.
The main contribution of this paper is a self-contained proof of an upper bound on the first- and second-order asymptotics of the parallel Gaussian channel with feedback. The proof techniques involve developing an information spectrum bound followed by using Curtiss' theorem to show that a sum of dependent random variables associated with the information spectrum bound converges in distribution to a sum of independent random variables, thus facilitating the use of the usual central limit theorem. Combined with existing achievability results, our result implies that the presence of feedback does not improve the first- and second-order asymptotics.
%\boldmath
\end{abstract}
% IEEEtran.cls defaults to using nonbold math in the Abstract.
% This preserves the distinction between vectors and scalars. However,
% if the journal you are submitting to favors bold math in the abstract,
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% of the abstract to achieve this. Many IEEE journals frown on math
% in the abstract anyway.

% Note that keywords are not normally used for peerreview papers.
\begin{IEEEkeywords}
%apacity region, cut-set outer bound, discrete memoryless network (AWGN channel), strong converse.
Curtiss' theorem, feedback, fixed-length codes, parallel Gaussian channel, second-order asymptotics
\end{IEEEkeywords}






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% \begin{center} \bfseries EDICS Category: 3-BBND \end{center}
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\section{Introduction} \label{Introduction}
This paper considers a point-to-point communication scenario where a source wants to transmit a message to a destination through a set of independent additive white Gaussian noise (AWGN) channels. The set of independent AWGN channels is referred to as the \emph{parallel Gaussian channel} \cite[Sec.~9.4]{CoverBook} (also called the \emph{Gaussian product channel} in~\cite[Sec.~3.4.3]{elgamal}). The parallel Gaussian channel has been used to model
%continuous-time AWGN channels where the parallel channels represent different orthogonal signal dimensions (such as frequency bands) and
the multiple-input multiple-output (MIMO) channel~\cite[Sec.~7.1]{davidTseBook} --- an essential channel model in wireless communications. Suppose the parallel Gaussian channel consists of~$L$ independent AWGN channels, and
let $\mathcal{L}\stackrel{\text{def}}{=} \{1, 2, \ldots, L\}$ be the index set of the $L$~channels.
% The message is uniformly distributed on a message alphabet of size~$M$.
For the $k^{\text{th}}$ channel use, the relation for the $\ell^{\text{th}}$ channel between the input signal $X_{\ell,k}$ and output signal $Y_{\ell,k}$ is
\begin{equation}
Y_{\ell,k} = X_{\ell,k} + Z_{\ell,k} \label{channelLawFormulation1Intro}
\end{equation}
where $\{Z_{\ell,k}\}_{\ell \in \mathcal{L}}$ are independent Gaussian noises. For each $\ell\in \mathcal{L}$, the variance of the noise induced by the $\ell^{\text{th}}$~channel is assumed to be some positive number $N_\ell>0$ for all channel uses, i.e., $\Var[Z_{\ell,k}]=N_\ell$ for all $k\in \mathbb{N}$.
%and the variance of $Z_{\ell,k}$ is assumed to be some positive number $N_\ell>0$.
 To keep notation compact, let $\boldsymbol{X}_k$, $\boldsymbol{Y}_k$ and $\boldsymbol{Z}_k$ denote the random column vectors $[X_{1, k}\ X_{2,k}\ \ldots \ X_{L,k}]^t$, $[Y_{1,k}\ Y_{2,k}\ \ldots \ Y_{L,k}]^t$ and $[Z_{1,k}\ Z_{2,k}\ \ldots \ Z_{L,k}]^t$ respectively. Then, the channel law~\eqref{channelLawFormulation1Intro} can be written as
\begin{equation}
\boldsymbol{Y}_k = \boldsymbol{X}_k + \boldsymbol{Z}_k. \label{channelLaw}
 \end{equation}
 %, and let $\mathbf{x}_k$, $\mathbf{y}_k$ and $\mathbf{z}_k$ be their realizations respectively.
%%Similarly, let $\boldsymbol{X}$ and $\boldsymbol{Y}$ denote the random column vectors $[X_1\ X_2\ \ldots \ X_L]^t$ and $[Y_1\ Y_2\ \ldots \ Y_L]^t$ respectively, and let $\mathbf{x}$ and $\mathbf{y}$ be their realizations respectively.
%Then, the channel law~\eqref{channelLawFormulation1Intro} can be written as
%\begin{equation}
%\boldsymbol{Y}_k = \boldsymbol{X}_k + \boldsymbol{Z}_k.
% \end{equation}
% We assume that a noiseless feedback link from node $\rm d$ to node $\rm s$ exists so that $(W, \boldsymbol{Y}^{k-1})$ is available for encoding $\boldsymbol{X}_k$ for each $k$.
Throughout this paper, we consider fixed-length codes over the parallel Gaussian channel, where the block length is denoted by~$n$ unless specified otherwise. Every codeword $\boldsymbol{X}^n$ transmitted by the source over~$n$ channel uses is subject to the following \emph{peak power constraint} where $P>0$ denotes the permissible power for~$\boldsymbol{X}^n$:
\begin{align}
\Pr\left\{\frac{1}{n}\sum_{\ell=1}^L \sum_{k=1}^n X_{\ell,k}^2 \le P\right\}=1.  \label{peakPowerConstraint}
\end{align}
%The additive white Gaussian noise (AWGN) channel is one in which at each discrete time $k \in \{1,2,\ldots, n\}$, the output $\boldsymbol{Y}_k$ is the sum of the input $\boldsymbol{X}_k$ and a Gaussian random variable $\boldsymbol{Z}_k$ that represents additive noise. The collection of the noise random variables $\{\boldsymbol{Z}_k\}_{k\in \{1,\ldots, n\}}$ is assumed to be independent and identically distributed (i.i.d.). The inputs are also power limited, which means that $\sum_{k=1}^n \boldsymbol{X}_k^2 \le nP$ with probability 1 where $P>0$ is the permissible power, i.e., a peak power constraint also called a maximum per-codeword power constraint.
 \indent
If we would like to transmit a uniformly distributed message $W\in \{1, 2, \ldots, \lceil 2^{nR} \rceil\}$ over this channel where the error probabilities are required to vanish as the blocklength~$n$ approaches infinity, it was shown by Shannon~\cite{Sha49} that the maximum rate of communication~$R$ converges to a certain limit called {\em capacity}. The closed-form expression of the capacity can be obtained by finding the optimal power allocation among the~$L$ channels, which is described as follows.
%The capacity of the parallel Gaussian  channel is well known and is
Define the mapping $\mathrm{C}(\mathbf{s}) : \mathbb{R}_+^L \rightarrow \mathbb{R}_+$ as
\begin{equation}
\mathrm{C}(\mathbf{s})= \sum_{\ell=1}^L \frac{1}{2}\log\left(1+\frac{s_\ell}{N_\ell}\right) \label{defCPparallelNFB}
\end{equation}
where $s_\ell$ can be viewed as the power allocated to channel~$\ell$.
 If we let $\Lambda$, $P_1$, $P_2$, $\ldots$, $P_L$ denote the $L+1$ real numbers yielded from the water-filling algorithm~\cite[Ch~9.4]{CoverBook} where
\begin{equation}
\sum_{\ell=1}^L P_\ell = P \label{sumPell=P}
\end{equation}
and
\begin{equation}
P_\ell = \max\{0,\Lambda - N_\ell\} \label{pEllValue}
\end{equation}
for each $\ell\in \mathcal{L}$ and let
\begin{equation}
\mathbf{P}^*\stackrel{\text{def}}{=}[P_1\ P_2\ \ldots \ P_L]^t \label{pEllValue*}
\end{equation}
be the optimal power allocation vector,
 then the capacity of the parallel Gaussian channel was shown in~\cite{Sha49} to be $\mathrm{C}(\mathbf{P}^*)$ bits per channel use.
%
%
% Recalling the definitions of $\mathrm{C}(\mathbf{P}^*)$ and $\mathrm{V}(P)$ in \eqref{defCPparallelNFB} and \eqref{defVP} respectively, we let
%
%and
%\begin{equation}
%\mathrm{V}(\mathbf{P}^*) \stackrel{\text{def}}{=} \sum_{\ell=1}^L \mathrm{V}(P_\ell/N_\ell) \qquad\mbox{bits$^2$ per channel use}, \label{defVPparallelNFB}
%\end{equation}
%be the capacity and the dispersion of the parallel Gaussian  channel \cite[Th.~78]{Pol10} respectively.
%
%\begin{equation}
%\mathrm{C}(\mathbf{P}^*) \stackrel{\text{def}}{=} \frac{1}{2}\log(1+P) \qquad\mbox{bits per channel use}. \label{defCP}
%\end{equation}
More specifically, if $M^*(n,\varepsilon,P)$ denotes the maximum number of messages that can be transmitted over $n$ channel uses with permissible power~$P$ and average error probability $\varepsilon$, one has
\begin{equation}
\lim_{\varepsilon\to 0}\liminf_{n\to\infty}\frac{1}{n}\log M^*(n,\varepsilon,P)=\mathrm{C}(\mathbf{P}^*). \label{shannonCapacity}
\end{equation}
%In fact, the strong converse was shown by Shannon in~\cite{Sha59b} (also see Yoshihara~\cite{Yoshihara} and Wolfowitz~\cite{Wolfowitz}) and   so we have
%\begin{equation*}
%\lim_{n\to\infty}\frac{1}{n}\log M^*(n,\varepsilon,P)=\mathrm{C}(\mathbf{P}^*)
%\end{equation*}
%for every $\varepsilon\in (0,1)$.
The capacity result~\eqref{shannonCapacity} has been strengthened by Polyanskiy-Poor-Verd\'u \cite[Th.~78]{Pol10} and Tan-Tomamichel \cite[Appendix~A]{TanTom13a} for each $\varepsilon\in(0,1)$ as
 \begin{equation}
\frac{1}{n}\log M^*(n,\varepsilon,P)=\mathrm{C}(\mathbf{P}^*) + \sqrt{\frac{\mathrm{V}(\mathbf{P}^*)}{n}}\, \Phi^{-1} (\varepsilon) + \Theta\Big(\frac{\log n}{n}\Big), \label{eqn:asymp_expans}
\end{equation}
where $\mathrm{V} : \mathbb{R}_+^L \rightarrow \mathbb{R}_+$ is
%follows in terms of the zstandard Bachmann-Landau notations (e.g., $O(\cdot)$, $o(\cdot)$, etc.),
the Gaussian dispersion function defined as
\begin{equation}
\mathrm{V}(\mathbf{s})=   \sum_{\ell=1}^L \frac{\frac{s_\ell}{N_\ell}(\frac{s_\ell}{N_\ell}+2) }{2(\frac{s_\ell}{N_\ell}+1)^2} \label{defVP}
\end{equation}
and $\Phi$ is the cumulative distribution function (cdf) of the standard normal distribution.
% Let  and $o(\cdot)$ s throughout the paper. Then, the optimal rate $\frac{1}{n}\log M^*(n,\varepsilon,P)$ admits the following asymptotic expansion for a fixed error probability~$\varepsilon$ \cite[Th.~78]{Pol10}\cite[Appendix~A]{TanTom13a} :
%where
%\begin{itemize}
%\item The mapping
%\item
%    \item
%\end{itemize}
%To facilitate discussion, we use $O(\cdot)$ and $o(\cdot)$ denote standard Bachmann-Landau notations. Define the mapping
%$\mathrm{V}(\mathbf{s}) : \mathbb{R}_+^L \rightarrow \mathbb{R}_+$ as
%
% investigated the optimal rate of a length-$n$ code under the setting that the average error probability of decoding the transmitted message is non-vanishing and showed that for each $\varepsilon\in(0,1)$, the optimal rate $\frac{1}{n}\log M^*(n,\varepsilon,P)$ admits the following asymptotic expansion:
%Around the same time, Hayashi~\cite{Hayashi09} provided another proof of \eqref{eqn:asymp_expans} with some third-order term $o(\sqrt{n})$.

{\em Feedback}, which is the focus of the current paper, can simplify coding schemes and improve the performance of communication systems in many scenarios. See \cite[Ch.~17]{elgamal} for a thorough discussion on the benefits of feedback in single- and multi-user information theory. When feedback is allowed, each input symbol $\boldsymbol{X}_k$ depends on not only the transmitted message $W$ but also all the previous channel outputs up to the $(k-1)^{\text{th}}$ channel use, i.e., the symbols $(\boldsymbol{Y}_1, \boldsymbol{Y}_2, \ldots, \boldsymbol{Y}_{k-1})$. %For {\em memoryless} AWGN channels,
In the presence of noiseless feedback, let $M_{\text{fb}}^*(n,\varepsilon,P)$ denote the maximum number of messages that can be transmitted over $n$ channel uses with permissible power~$P$ and average error probability $\varepsilon$.
It was shown by Shannon~\cite{Sha56} that the presence of noiseless feedback does not increase the capacity of point-to-point \emph{memoryless channels}, which together with~\eqref{shannonCapacity} implies that
\begin{align}
\lim_{\varepsilon\to 0}\liminf_{n\to\infty}\frac{1}{n}\log M_{\text{fb}}^*(n,\varepsilon,P)=\mathrm{C}(\mathbf{P}^*). \label{shannonFBCapacity}
\end{align}
In view of~\eqref{eqn:asymp_expans}, we conclude that
%the optimal rate $\frac{1}{n}\log M_{\text{fb}}^*(n,\varepsilon,P)$ satisfies
\begin{equation}
\frac{1}{n}\log M_{\text{fb}}^*(n,\varepsilon,P)\ge \mathrm{C}(\mathbf{P}^*) + \sqrt{\frac{\mathrm{V}(\mathbf{P}^*)}{n}}\, \Phi^{-1} (\varepsilon) + \Theta\Big(\frac{\log n}{n}\Big). \label{eqn:asymp_expans*}
\end{equation}
%It was shown by Shannon~\cite{Sha56} that the presence of noiseless feedback does not increase the capacity of point-to-point \emph{memoryless channels}. Therefore, the feedback capacity of the parallel Gaussian channel remains to be $\mathrm{C}(\mathbf{P}^*)$.
%%In this paper, we are interested in analyzing the performance of the parallel Gaussian channel with feedback under the constraint that the average error probability of decoding the transmitted message is non-vanishing and bounded above by a constant $\varepsilon\in (0,1)$.
%In the presence of feedback, if we let $M_{\text{fb}}^*(n,\varepsilon,P)$ denote the maximum number of messages that can be transmitted over $n$ channel uses with permissible power~$P$ and average error probability $\varepsilon$, it follows directly from~\eqref{eqn:asymp_expans} that the optimal rate $\frac{1}{n}\log M_{\text{fb}}^*(n,\varepsilon,P)$ satisfies
%\begin{equation}
%\frac{1}{n}\log M_{\text{fb}}^*(n,\varepsilon,P)\ge \mathrm{C}(\mathbf{P}^*) + \sqrt{\frac{\mathrm{V}(\mathbf{P}^*)}{n}}\, \Phi^{-1} (\varepsilon) + \Theta\Big(\frac{\log n}{n}\Big). \label{eqn:asymp_expans*}
%\end{equation}
%where
%\begin{equation}
%\mathrm{V}(P) \stackrel{\text{def}}{=}  \frac{P(P+2) }{2(P+1)^2}\qquad\mbox{bits}^2 \mbox{ per channel use} \label{defVP}
%\end{equation}
%is known as the Gaussian {\em dispersion} function and $\Phi^{-1}$ is the inverse of the cumulative distribution function for the standard Gaussian distribution. See Hayashi's work~\cite{Hayashi09} for a proof of \eqref{eqn:asymp_expans} without the  third-order $\frac{1}{2}\log n +O(1)$ term.
%
%\subsection{Main Contributions}
%%A natural question then arises. In the presence of feedback, what is the analogue of the asymptotic expansion in \eqref{eqn:asymp_expans}?
%Let $M_{\mathrm{fb}}^*(n,\varepsilon,P)$ be the maximum number of codewords that can be transmitted through $n$ uses of the channel when each input symbol $\boldsymbol{X}_k$ is allowed to depend on $(W,\boldsymbol{Y}^{k-1})$. Clearly, $M_{\mathrm{fb}}^*(n,\varepsilon,P)\ge M^*(n,\varepsilon,P)$ for all choices of the parameters $(n,\varepsilon,P)$ (because the code can simply ignore the fed back symbols $Y^{k-1}$).
In this paper, the main contribution is a conceptually simple, concise and self-contained proof that in the presence of feedback, the first- and second-order terms in the asymptotic expansion in~\eqref{eqn:asymp_expans} remains unchanged, i.e.,
\begin{equation}
\frac{1}{n}\log M_{\mathrm{fb}}^*(n,\varepsilon,P)=\mathrm{C}(\mathbf{P}^*) + \sqrt{\frac{\mathrm{V}(\mathbf{P}^*)}{n}}\, \Phi^{-1} (\varepsilon) + o\Big(\frac{1}{\sqrt{n}}\Big).\label{eqn:asymp_expans_fb}
\end{equation}
%where $o(\cdot)$ denotes the standard Bachmann-Landau notation.
%This means that, up to the third-order term in the asymptotic expansion of $\log M_{\mathrm{fb}}^*(n,\varepsilon,P)$, full feedback from the output of the channel to the encoder does not increase the number of codewords transmissible over the channel.

%This is somewhat surprising  (at least to the authors) given that the error probability performance improves greatly in the presence of feedback for fixed rates below capacity~\cite{Schalkwijk,Schalkwijk1,  Shayevitz}.

%As an auxiliary contribution, we investigate the parallel Gaussian channel with feedback under a peak power constraint and prove an upper bound for the second-order asymptotics. This establishes the strong converse for this channel, which (to the best of the authors' knowledge) was not known previously.
%\subsection{Main Contribution}


\subsection{Related Work} \label{subsecRelatedWork}
Our work is inspired by the recent study of the fundamental limits of communication over discrete memoryless channels (DMCs) with feedback~\cite{AW14}. It was shown by Altu\u{g} and Wagner~\cite[Th.~1]{AW14} that for some classes of DMCs whose capacity-achieving input distributions are not unique (in particular, the minimum and maximum conditional information variances differ), coding schemes with feedback achieve a better second-order asymptotics compared to those without feedback. They also showed \cite[Th.~2]{AW14} that feedback does not improve the second-order asymptotics of DMCs $q_{Y|X}$ if the conditional variance of the log-likelihood ratio $\log \frac{q_{Y|X}(Y|x)}{p^*(Y)}$, where $p^*$ is the unique capacity-achieving output distribution, does not depend on the input $x$. Such DMCs include the class of weakly-input symmetric DMCs initially studied by Polyanskiy-Poor-Verd\'u~\cite{PPV11b}.
%Our contribution is similar in spirit to \cite[Th.~2]{AW14}.

However, we note that the proof technique used by Altu\u{g} and Wagner requires the use of a Berry-Ess\'een-type result for bounded martingale difference sequences~\cite{Machkouri}, and their technique cannot be extended to the parallel Gaussian channel with feedback because each input symbol~$X_{\ell,k}$ belongs to an interval $[-\sqrt{nP}, \sqrt{nP}]$ that grows without bound as~$n$ increases. Instead, our proof uses Curtiss' theorem to show that a sum of dependent random variables that naturally appears in the non-asymptotic analysis converges in distribution to a sum of independent random variables, thus facilitating the use of the usual central limit theorem~\cite{feller}.  %we partition transmitted codewords %according to their power types and apply Curtiss' theorem for the codewords whose power types converge to the optimal power allocation. In additoin, for the codewords whose power types are far from the optimal power %allocation, we apply large deviation bounds

For $L=1$, the parallel Gaussian channel with feedback reduces to the AWGN channel with feedback, whose second-order coding rate is identical to the same channel without feedback by the following symmetry argument: The log-likelihood ratios $\log \frac{q_{Y|X}(Y|x)}{p^*(Y)}$ for all $x$ on the power sphere with radius $\sqrt{nP}$ are the same. See~\cite{FongTan15} for a rigorous but simple proof. In contrast, for $L>1$, this symmetry argument no longer holds due to the flexible power allocation among the~$L$ channels, and hence the simple proof suggested in~\cite{FongTan15} cannot be extended to the parallel Gaussian channel with feedback.

If the peak power constraint in~\eqref{peakPowerConstraint} is replaced with the expected power constraint $\E\left[\frac{1}{n}\sum_{\ell=1}^L \sum_{k=1}^n X_{\ell,k}^2 \right]\le P$, the first-order coding rate of the AWGN channel with feedback is improved from $\mathrm{C}(P)$ to $\mathrm{C}(\frac{P}{1-\varepsilon})$~\cite[Sec.~II]{TFT17} (the exact improvement holds for the non-feedback case as well~\cite[Sec.\ 4.3.3]{Pol10}) where $\varepsilon$ denotes the tolerable error probability. For the general case $L>1$, the proof in \cite[Sec.~II]{TFT17} can be easily extended to show that the first-order coding rate of the parallel Gaussian channel with feedback can be improved from $\mathrm{C}(\mathbf{P}^*)$ to $\mathrm{C}(\frac{\mathbf{P}^*}{1-\varepsilon})$, and hence~\eqref{eqn:asymp_expans_fb} no longer holds.
%Our technique for the AWGN channel is conceptually simpler. We prove that a sum of random variables that naturally appears in the non-asymptotic analysis of the AWGN channel with feedback has the same distribution as the sum of i.i.d.\ random variables, thus facilitating the use of the usual Berry-Ess\'een theorem~\cite[Th.~2 in Section XVI.5]{feller}. We prove this equivalence between the distributions by using moment generating functions.

%Our result in  \eqref{eqn:asymp_expans_fb} is somewhat expected in light of the fact that
%In another line of work, for rates below capacity $\mathrm{C}(P)$, Pinsker~\cite{Pin68} showed that for fixed-length block codes on Gaussian channels, the use of feedback cannot improve the exponent over the sphere-packing bound under the peak power constraint. This is in contrast to the case where an expected average power constraint is imposed under which Schalkwijk and Kailath \cite{Schalkwijk,Schalkwijk1} showed in a series of celebrated works that the error probability decays doubly exponentially fast for rates below $\mathrm{C}(P)$. The work contained herein {\em only} considers the peak power constraint. We leave the analysis of the asymptotic expansion for $\log M_{\mathrm{fb}}^*(n,\varepsilon,P)$ under the expected average power constraint for future work.

%In another line of work, Schalkwijk and Kailath \cite{Schalkwijk} and Schalkwijk \cite{Schalkwijk1} showed for the AWGN channel with feedback that for fixed rates below capacity, the error probability decays doubly exponentially fast. This coding strategy now known as {\em posterior matching} has been and studied more extensively for a much wider class of channels by Shayevitz and Feder~\cite{Shayevitz}. It has also been showed by Pinsker \cite{Pin68}, Zigangirov~\cite{Zigangirov70} and Kramer~\cite{Kramer69} that the probability of error can be made to decay as fast as an arbitrary level of nested exponentials.  The fact that the asymptotic expansion of $\log M^*(n,\varepsilon,P)$  (up to the third-order term) is unaffected by the presence of feedback lies in stark contrast to posterior matching in which the error exponent is infinite for rates below capacity.

%For the parallel AWGN channel with feedback, we could not characterize the second- and third-order asymptotics due to an obstacle to be addressed in the last section of this paper. Instead, we use Chebyshev's inequality
%to prove the strong converse for the parallel AWGN channel with feedback.

\subsection{Paper Outline}
This paper is organized as follows. The next subsection summarizes the notation used in this paper. Section~\ref{sectionDefinition} provides the problem setup of the parallel Gaussian channel with feedback under the peak power constraint and presents our main theorem. Section~\ref{sectionPrelim} contains the preliminaries required for the proof of our main theorem. The preliminaries include the following: (i) Important properties of non-asymptotic binary hypothesis testing quantities; (ii) Modification of power allocation among the parallel channels; (iii) Curtiss' theorem. Section~\ref{sectionMainResult} presents the proof of our main theorem.
Section~\ref{sectionConclusion} concludes this paper by explaining the novel ingredients in the proof of the main theorem and the major difficulty in strengthening the main theorem.
%Section~\ref{sectionNovelIngredients}  explains the novel ingredients in the proof of the main theorem.
%Section~\ref{sectionParallelAWGNChannel} discusses the parallel Gaussian channel with feedback under a peak power constraint and applies the techniques used in Section~\ref{sectionMainResult} to prove the strong converse.
%We conclude our discussion and suggest avenues for future research in Section~\ref{sec:conclu}.
\subsection{Notation}\label{notation}
The sets of natural numbers, non-negative integers, real numbers and non-negative real numbers are denoted by $\mathbb{N}$, $\mathbb{Z}_+$, $\mathbb{R}$ and $\mathbb{R}_+$ respectively.
 %The $L$-dimensional identity matrix is denoted by $I_L$, the length-$L$ all-zero column vector is denoted by $\mathbf{0}_L$, the $L_1\times L_2$ all-zero matrix is denoted by $0^{L_1\times L_2}$, and the $L_1\times L_2$ all-one matrix is denoted by $1^{L_1\times L_2}$. For any real-valued matrix $K$, we let $K^t$ denote its transpose. If $K$ is a square matrix, we use $K\succ 0$ and $K\succeq 0$ to represent that $K$ is positive definite and $K$ is positive semi-definite respectively, and let $\diag(K)$, $|K|$ and $\Tr(K)$ denote the diagonal column vector, determinant and trace of $K$ respectively.
 %We let $K^{-1}$ denote the inverse of any invertible matrix~$K$.
 %For any two real-valued matrices $A$ and $B$ of the same dimension, we use $A<B$, $A\le B$ and $A=B$ to represent the corresponding relations between $A$ and $B$ entrywise.
 An $L$-dimensional column vector is denoted by $\mathbf{a} \stackrel{\text{def}}{=} [a_1\ a_2\ \ldots \ a_L]^t$ where $a_\ell$ denote the $\ell^{\text{th}}$ element of $\mathbf{a}$. The Euclidean norm of a vector $\mathbf{a}\in \mathbb{R}^L$ is denoted by %$\mathcal{L}_1$-distance between
%two distributions $p_X$ and $q_X$ on the same alphabet $\mathcal{X}$, denoted by
$\|\mathbf{a}\|_{2}\stackrel{\text{def}}{=} \sqrt{\sum_{\ell=1}^L a_\ell^2}$\,. We will take all logarithms to base~$e$ throughout this paper.

We use $\Pr\{\mathcal{E}\}$ to represent the probability of an
event~$\mathcal{E}$, and we let $\mathbf{1}\{\mathcal{E}\}$ be the indicator function of $\mathcal{E}$. Every random variable is denoted by a capital letter (e.g., $X$), and the realization and the alphabet of the random variable are denoted by the corresponding small letter (e.g., $x$) and calligraphic letter (e.g., $\mathcal{X}$) respectively.
We use $X^n$ to denote a random tuple $(X_1,  X_2,  \ldots ,  X_n)$, where all the elements $X_k$ have the same alphabet~$\mathcal{X}$. We let $p_X$ be the probability distribution of a random variable $X$. More specifically, $p_X$ is the Radon-Nikodym derivative of a measure with respect to the Lebesgue measure in an appropriate Euclidean space. We let $p_{Y|X}$ denote the conditional probability distribution of $Y$ given $X$ for any random variables~$X$ and~$Y$.
%For any mapping $g$ whose domain includes $\mathcal{X}$, we let $p_{g(X)}$ denote the probability mass distribution of $g(X)$ when $X$ is distributed according to $p_X$.
%We let $p_X(x)\stackrel{\text{def}}{=}\Pr\{X=x\}$ and $p_{Y|X}(y|x)\stackrel{\text{def}}{=}\Pr\{Y=y|X=x\}$ be the evaluations of $p_X$ and $p_{Y|X}$ respectively at $X=x$ and $Y=y$.
We let $p_Xp_{Y|X}$ denote the joint distribution of $(X,Y)$, i.e., $p_Xp_{Y|X}(x,y)=p_X(x)p_{Y|X}(y|x)$ for all $x$ and $y$.
%If $X$ and $Y$ are independent, their joint distribution is simply $p_X p_Y$.
 For any random variable~$X\sim p_X$ and any real-valued function~$g$ whose domain includes $\mathcal{X}$, we let $\Pr_{p_X}\{g(X)\ge\xi\}$ denote $\int_{\mathcal{X}} p_X(x)\mathbf{1}\{g(x)\ge\xi\}\, \mathrm{d}x$ for any real constant $\xi$ where $p_X$  The expectation and the variance of~$g(X)$ are denoted as
$
\E_{p_X}[g(X)]$ and
$
 \Var_{p_X}[g(X)]$ respectively.
%For any function $f$ whose domain contains $\mathcal{X}$, we use $\E_{p_X}[f(X)]$ and $\Var_{p_X}[f(X)]$ to denote the expectation and variance of~$f(X)$ respectively where $X$ is distributed according to $p_X$.
%The $L\times L$ covariance matrix of the random column vector~$\boldsymbol{X}\stackrel{\text{def}}{=}[X_1 \ X_2\ \ldots X_L]^t\sim p_{\boldsymbol{X}}$ is denoted by $\Cov_{p_{\boldsymbol{X}}}[\boldsymbol{X}]$.
%For any function $f$ whose domain contains $\mathcal{Y}$, we use $\E_{p_X(y)}[f(Y)]\stackrel{\text{def}}{=} \sum_{y\in\mathcal{Y}}f(y) p_X(y)$ to represent the expectation of a discrete random variable~$f(Y)$ where $Y$ is distributed according to $p_X$.
%We use $x_{(1, \ldots, L)}$ and $x_{(1, \ldots, L)}^n$ to denote the realizations of~$X_{(1, \ldots, L)}$ and $X_{(1, \ldots, L)}^n$ respectively.
 For simplicity, we drop the subscript of a notation if there is no ambiguity.
 %For any random tuple $(U, X,Y,Z)$ distributed according to $p_{U, X,Y,Z}$, we let $h_{p_{U,X,Y,Z}}(X|Z)$ or more simply $h_{p_{X,Z}}(X|Z)$ denote the differential entropy of $X$ given $Z$, and let $I_{p_{U, X,Y,Z}}(X;Y|Z)$ or more simply $I_{p_{X,Y,Z}}(X;Y|Z)$ denote the mutual information between $X$ and $Y$ given~$Z$.
 For any real-valued Gaussian random variable $Z$ whose mean and variance are $\mu$ and $\sigma^2$ respectively, we let
 \begin{equation}
 \mathcal{N}(z; \mu, \sigma^2)\stackrel{\text{def}}{=} \frac{1}{\sqrt{2\pi \sigma^2}}e^{-\frac{(z-\mu)^2}{2\sigma^2}} \label{normalDist}
 \end{equation}
 be the corresponding probability density function.
% For any $L$-dimensional real-valued Gaussian vector $\boldsymbol{Z}\stackrel{\text{def}}{=} [Z_1\ Z_2\ \ldots\ Z_L]^t$ whose mean and covariance are $\boldsymbol{\mu}\in \mathbb{R}^L$ and~$\Sigma\in \mathbb{R}^{L\times L}$ respectively, we let
% \begin{equation}
% \mathcal{N}(\mathbf{z}; \boldsymbol{\mu}, \Sigma)\stackrel{\text{def}}{=} \frac{1}{\sqrt{(2\pi)^L |\Sigma|}}e^{-\frac{1}{2}(\mathbf{z}-\boldsymbol{\mu})^t \Sigma^{-1} (\mathbf{z}-\boldsymbol{\mu}) } \label{normalDist}
% \end{equation}
% be the corresponding probability density function.

%
%
%We use $\Pr\{\mathcal{E}\}$ to represent the probability of an
%event~$\mathcal{E}$, and we let $\boldsymbol{1}(\mathcal{E})$ be the indicator function of $\mathcal{E}$. We use a capital letter~$X$ to denote an arbitrary (can be discrete, continuous or other general) random variable with alphabet $\mathcal{X}$, and use the small letter $x$ to denote a realization of~$X$.
%%We use $X_{(1, \ldots, N)}$ to denote a random tuple $(X_1, X_2, \ldots, X_N)$, where each element $\boldsymbol{X}_k$ is chosen from some alphabet $\mathcal{X}_k$ for $k=1, 2, \ldots, N$.
%We use $X^n$ to denote a random tuple $(X_1, X_2, \ldots, X_n)$, where the components $\boldsymbol{X}_k$ have the same alphabet~$\mathcal{X}$.
%
%The following notations are used for any arbitrary random variables~$X$ and~$Y$ and any mapping $g$ whose domain includes $\mathcal{X}$. We let $p_X$ and $p_{Y|X}$ denote the probability distribution of $X$ and the conditional probability distribution of $Y$ given $X$ respectively.
%%We let $p_X(x)$ and $p_{Y|X}(y|x)$ be the evaluations of $p_X$ and $p_{Y|X}$ respectively at $X=x$ and $Y=y$. We let $p_{g(X)}$ denote the probability distribution of $g(X)$ when $X$ is distributed according to $p_X$, and
%We let $\Pr_{p_X}\{g(X)\ge\xi\}$ denote $\int_{x\in \mathcal{X}} p_X(x)\mathbf{1}(\{g(x)\ge\xi\})\, \mathrm{d}x$ for any real-valued function~$g$ and any real constant $\xi$. The expectation and the variance of~$g(X)$ are denoted as
%$
%\E_{p_X}[g(X)]$ and
%$
% \Var_{p_X}[g(X)]\stackrel{\text{def}}{=} \E_{p_X}[(g(X)-\E_{p_X}[g(X)])^2]$
% respectively.
%  We let $p_Xp_{Y|X}$ denote the joint distribution of $(X,Y)$, i.e., $p_Xp_{Y|X}(x,y)=p_X(x)p_{Y|X}(y|x)$ for all $x$ and $y$.
%  %If $X$ and $Y$ are independent, their joint distribution is simply $p_X p_Y$.
%  We let $\phi_{\mu,\sigma^2}: \mathbb{R}\rightarrow [0,\infty)$ denote the probability density function of a Gaussian random variable whose mean and variance are $\mu$ and $\sigma^2$ respectively such that
%  \[
%  \phi_{\mu,\sigma^2}(z) = \frac{1}{\sqrt{2\pi \sigma^2}}e^{-\frac{(z-\mu)^2}{2\sigma^2}}.
%  \]
%  We will take all logarithms to base 2 throughout this paper.
   %The set of natural numbers and real numbers are denoted by $\mathbb{N}$ and $\mathbb{R}$ respectively.
%The following notations are used for any arbitrary random variables~$X$ and~$Y$ and any mapping $g$ whose domain includes $\mathcal{X}$. We let $f_X$ and $f_{Y|X}$ denote the probability density distribution of $X$ and the conditional probability density distribution of $Y$ given $X$ respectively. We define $f_X(x)$, $f_{Y|X}(y|x)$, $f_Xf_{Y|X}$, $\E_{p_X}[g(X)]$ and $\Var_{p_X}[X]$ similar to the discrete case above.
%For any function $f$ whose domain contains $\mathcal{Y}$, we use $\E_{p_X(y)}[f(Y)]\stackrel{\text{def}}{=} \sum_{y\in\mathcal{Y}}f(y) p_X(y)$ to represent the expectation of a discrete random variable~$f(Y)$ where $Y$ is distributed according to $p_X$.
%We use $x_{(1, \ldots, N)}$ and $x_{(1, \ldots, N)}^n$ to denote the realizations of~$X_{(1, \ldots, N)}$ and $X_{(1, \ldots, N)}^n$ respectively.
% For simplicity, we drop the subscript of a notation if there is no ambiguity.

 %For any arbitrary random variables $X$, $Y$, and $Z$ distributed according to $p_{X,Y,Z}$ and any mapping $g$ whose domain includes $\mathcal{X}$, we let $p_{X|Y, Z=z}$ be the conditional probability measure (distribution) of $X$ given $Y$ under the event $\{Z=z\}$, and we let
% \begin{align*}
% \E_{p_{X|Y, Z=z}}[g(X)|y] & \stackrel{\text{def}}{=} \int_{x\in \mathcal{X}} p_{X|Y,Z=z}(x|y)g(x) \, \mathrm{d}x \notag\\
% &=\int_{x\in \mathcal{X}} p_{X|Y,Z}(x|y,z)g(x) \, \mathrm{d}x
% \end{align*}
% be the conditional expectation of $g(X)$ given $\{Y=y\}$ under the event $\{Z=z\}$.
 %Viewing $\E_{p_{X|Y, Z=z}}[g(X)|y]$ as a function of $y$, we regard the expression
%\[
%\E_{p_{X|Y, Z=z}}[g(X)|Y]
%\]
%as a random variable which is a function of $Y$. We use
% \[
% H_{p_{X,Z}}(X|Z)\stackrel{\text{def}}{=} \E_{p_{X,Z}}\left[\log\frac{1}{p_{X|Z}(X|Z)}\right]
%  \]
%  and
%  \[
%  I_{p_{X,Y,Z}}(X;Y|Z)\stackrel{\text{def}}{=} \E_{p_{X,Y,Z}}\left[\log\frac{p_{X,Y|Z}(X,Y|Z)}{p_{X|Z}(X|Z)p_{Y|Z}(Y|Z)}\right]
%   \]
%to denote the entropy of $X$ given $Z$ and mutual information between $X$ and $Y$ given $Z$ respectively.

 %The $\mathcal{L}_1$-distance between
%two distributions $p_X$ and $q_X$ on the same alphabet $\mathcal{X}$, denoted by $\|p_X-q_X\|_{\mathcal{L}_1}$, is defined as $\|p_X-q_X\|_{\mathcal{L}_1}\stackrel{\text{def}}{=} \int_{x\in\mathcal{X}}|p_X(x)-q_X(x)|\, \mathrm{d}x$.
 %denote $\Pr\{X=x\}$ and $\Pr\{Y=y|X=x\}$ respectively.
 %If $X$, $Y$ and $Z$ are distributed according to $p_{X,Y,Z}$ and they form a Markov chain, we write
%$(X\rightarrow Y\rightarrow Z)_{p_{X,Y,Z}}$ or more simply, $(X\rightarrow Y\rightarrow Z)_p$.

%For any set~$S$, we let $S^n$ to be the Cartesian product of $n$ copies of~$S$.


\section{Parallel Gaussian Channel with Feedback} \label{sectionDefinition}
 %communicating a message from one source denoted by $\mathrm{s}$ to one destination denoted by $\mathrm{d}$ through a set of
% We consider the scenario where a source denoted by $\mathrm{s}$ wants to transmit a message to a destination denoted by $\mathrm{d}$ through a set of independent additive white Gaussian noise (AWGN) channels, which is referred to as parallel Gaussian channel \cite[Section~9.4]{CoverBook}. More specifically,
 Let~$\mathrm{s}$ and~$\mathrm{d}$ denote the source and the destination respectively. Suppose node~$\mathrm{s}$ transmits a message to node~$\mathrm{d}$ over $n$ channel uses through the $L$~independent AWGN channels.
 %Let $\mathcal{L}\stackrel{\text{def}}{=} \{1, 2, \ldots, L\}$ be the index set of the $L$~channels.
 Before any transmission begins, node~$\mathrm{s}$ chooses message~$W$ destined for node~$\mathrm{d}$ where $W$ is uniformly distributed on the message alphabet
\begin{equation}
\mathcal{W}\stackrel{\text{def}}{=} \{1, 2, \ldots, M\} \label{defMessageAlphabet}
 \end{equation}
whose size is denoted by $M$. %For each $k\in\{1, 2, \ldots, n\}$ and each $\ell\in \mathcal{L}$, the channel law is described as follows:
For the $k^{\text{th}}$ channel use, node~$\mathrm{s}$ transmits $\boldsymbol{X}_{k}$ and the corresponding channel output~$\boldsymbol{Y}_{k}$ satisfies~\eqref{channelLaw}.
 We assume that a noiseless feedback link from the destination node $\rm d$ to the source node $\rm s$ exists so that $(W, \boldsymbol{Y}^{k-1})$ is available for encoding $\boldsymbol{X}_k$ for each $k\in\{1, 2, \ldots, n\}$. In addition, the codewords $\boldsymbol{X}^n$ transmitted by~$\mathrm{s}$ is subject to the peak power constraint~\eqref{peakPowerConstraint}. Upon receiving $\boldsymbol{Y}^n$, node~$\mathrm{d}$ declares $\hat W$ to be the transmitted message.
%  where $P>0$ denotes the permissible power for $\boldsymbol{X}^n$. In other words,
%\begin{align}
%\Pr\left\{\sum_{\ell=1}^L \sum_{k=1}^n X_{\ell,k}^2 \le n P\right\}=1.  \label{peakPowerConstraint}
%\end{align}
\medskip
\begin{Definition} \label{defCode}
An $(n, M, P)$-feedback code consists of the
following:
\begin{enumerate}
\item A message set $\mathcal{W}$ at node~$\mathrm{s}$ as defined in~\eqref{defMessageAlphabet}. Message $W$ is uniform on $\mathcal{W}$.

\item An encoding function
\[
f_{\ell,k} : \mathcal{W}\times \mathbb{R}^{L\times (k-1)}\rightarrow \mathbb{R}
 \]
 for each $\ell\in\mathcal{L}$ and each $k\in\{1, 2, \ldots, n\}$, where $f_{\ell,k}$ is the encoding function at node~$\mathrm{s}$ for encoding $X_{\ell,k}$ such that
\[
X_{\ell,k}=f_{\ell,k} (W, \boldsymbol{Y}^{k-1})
\]
and the peak power constraint~\eqref{peakPowerConstraint} holds.
\item A decoding function
\[
\varphi :
\mathbb{R}^{L\times n} \rightarrow \mathcal{W},
 \]
where $\varphi$ is the decoding function for $W$ at node~$  \mathrm{d}$ such that
 \[
 \hat W = \varphi(\boldsymbol{Y}^{n}).
 \]
\end{enumerate}
\end{Definition}
\medskip
\begin{Definition}\label{defAWGNchannel}
 Let $\boldsymbol{X}$ and $\boldsymbol{Y}$ denote the random vectors $[X_{1}\ X_{2}\ \ldots \ X_{L}]^t$ and $[Y_{1}\ Y_{2}\ \ldots \ Y_{L}]^t$ respectively, and let $\mathbf{x}$ and $\mathbf{y}$ be their realizations respectively. The \emph{parallel Gaussian channel with feedback} is characterized by the conditional probability density distribution $q_{\boldsymbol{Y}|\boldsymbol{X}}$ satisfying
\begin{equation}
q_{\boldsymbol{Y}|\boldsymbol{X}}(\mathbf{y}|\mathbf{x})= \prod_{\ell=1}^L \mathcal{N}(y_\ell; x_\ell, N_\ell) \label{defAWGNchannelParallel}
\end{equation}
such that the following holds for any $(n, M, P)$-feedback code: For each $k\in\{1, 2, \ldots, n\}$,
\begin{align}
p_{W, \boldsymbol{X}^k, \boldsymbol{Y}^k}
 = p_{W, \boldsymbol{X}^k, \boldsymbol{Y}^{k-1}}p_{\boldsymbol{Y}_k|\boldsymbol{X}_k} \label{memorylessStatement*}
\end{align}
where
\begin{equation}
p_{\boldsymbol{Y}_k|\boldsymbol{X}_k}(\mathbf{y}_k|\mathbf{x}_k) = q_{\boldsymbol{Y}|\boldsymbol{X}}(\mathbf{y}_k|\mathbf{x}_k) \label{defChannelInDefinition*}
\end{equation}
for all $(\mathbf{x}^n,\mathbf{y}^n)\in \mathbb{R}^{L\times n}\times\mathbb{R}^{L\times n}$.
%Since $p_{\boldsymbol{Y}_k|\boldsymbol{X}_k}$ does not depend on~$k$ by \eqref{defChannelInDefinition*}, the channel is stationary.
\end{Definition}
\medskip

 For any $(n, M, P)$-feedback code, let $p_{W,\boldsymbol{X}^n, \boldsymbol{Y}^n, \hat W}$ be the joint distribution induced by the code. We can use Definition~\ref{defCode}, \eqref{memorylessStatement*} and \eqref{defChannelInDefinition*} to factorize $p_{W,\boldsymbol{X}^n, \boldsymbol{Y}^n, \hat W}$ as follows:
\begin{align}
 p_{W,\boldsymbol{X}^n, \boldsymbol{Y}^n, \hat W}
%&\stackrel{\text{(a)}}{=} p_{W,\boldsymbol{X}^n, \boldsymbol{Y}^n}p_{\hat W |\boldsymbol{Y}^n}  \notag\\
%&= p_W \left(\prod_{k=1}^np_{\boldsymbol{X}_k,\boldsymbol{Y}_k|\boldsymbol{X}^{k-1}, \boldsymbol{Y}^{k-1}, W}\right)p_{\hat W |\boldsymbol{Y}^n} \notag\\
%&= p_W \left(\prod_{k=1}^n p_{\boldsymbol{X}_k|\boldsymbol{X}^{k-1}, \boldsymbol{Y}^{k-1}, W}p_{\boldsymbol{Y}_k|\boldsymbol{X}^k, \boldsymbol{Y}^{k-1}, W} \right)p_{\hat W |\boldsymbol{Y}^n} \notag\\
%& \stackrel{\text{(b)}}{=} p_W \left(\prod_{k=1}^n \left( p_{\boldsymbol{X}_k|W, \boldsymbol{Y}^{k-1}} p_{\boldsymbol{Y}_k|\boldsymbol{X}^k, \boldsymbol{Y}^{k-1}, W} \right)\right)p_{\hat W |\boldsymbol{Y}^n}\notag\\
& = p_W \left(\prod_{k=1}^n p_{\boldsymbol{X}_k|W, \boldsymbol{Y}^{k-1}} p_{\boldsymbol{Y}_k|\boldsymbol{X}_k}\right)p_{\hat W |\boldsymbol{Y}^n}. \label{memorylessStatement}
\end{align}
%where
%\begin{enumerate}
%\item[(a)] follows from Definition~\ref{defCode} that $\hat W$ is a function of $Y^n$.
%\item[(b)] follows from Definition~\ref{defCode} that $\boldsymbol{X}_k$ is a function of $(W, \boldsymbol{Y}^{k-1})$ for each $k\in\{1, 2, \ldots, n\}$.
%   \item[(c)] follows from \eqref{memorylessStatement*} and \eqref{defChannelInDefinition*} that for all $w$, $x^k$ and $y^k$ such that $p_{X^k, \boldsymbol{Y}^{k-1}, W}(x^k, y^{k-1}, w)>0$,
%   \begin{equation}
%    p_{\boldsymbol{Y}_k|W, X^k, \boldsymbol{Y}^{k-1}}(y_k|w, x^k, y^{k-1}) = p_{\boldsymbol{Y}_k|\boldsymbol{X}_k}(y_k|x_k) = q_{Y|X}(y_k|x_k). \label{memorylessStatement+}
%    \end{equation}
%\end{enumerate}
 % It can be derived from Definition~\ref{defAWGNchannel} that $\{\boldsymbol{Y}_k-\boldsymbol{X}_k\}_{k=1}^n$ are independent standard Gaussian random variables, which is shown below for completeness. For each $k\in\{1, 2, \ldots, n\}$,
% \begin{align*}
%& p_{W, X^k, \boldsymbol{Y}^{k-1}, \boldsymbol{Y}_k-\boldsymbol{X}_k}(w, x^k, y^{k-1}, y_k-x_k) \notag\\
%& = p_{W, X^k, \boldsymbol{Y}^{k-1}}(w, x^k, y^{k-1}) p_{\boldsymbol{Y}_k-\boldsymbol{X}_k|W, X^k, \boldsymbol{Y}^{k-1}}(y_k-x_k|w, x^k, y^{k-1}) \notag\\
%& = p_{W, X^k, \boldsymbol{Y}^{k-1}}(w, x^k, y^{k-1}) p_{\boldsymbol{Y}_k|W, X^k, \boldsymbol{Y}^{k-1}}(y_k|w, x^k, y^{k-1}) \notag\\
%&\stackrel{\eqref{memorylessStatement*}}{=}p_{W, X^k, \boldsymbol{Y}^{k-1}}(w, x^k, y^{k-1})q_{\boldsymbol{Y}_k|\boldsymbol{X}_k}(y_k|x_k)\notag\\
%&  \stackrel{\eqref{defChannelInDefinition}}{=}p_{W, X^k, \boldsymbol{Y}^{k-1}}(w, x^k, y^{k-1})\phi_{(\boldsymbol{Y}_k-\boldsymbol{X}_k; 0,1)}(y_k-x_k)
% \end{align*}
% for all $w$, $x^k$ and $y^k$, which implies that
% \begin{equation}
% p_{\boldsymbol{Y}_k-\boldsymbol{X}_k|W, X^k, \boldsymbol{Y}^{k-1}} = p_{\boldsymbol{Y}_k-\boldsymbol{X}_k} = \phi_{(\boldsymbol{Y}_k-\boldsymbol{X}_k; 0,1)}. \label{distributionOfYk-Xk}
% \end{equation}
% Consequently,
% \begin{align}
% p_{\{\boldsymbol{Y}_k-\boldsymbol{X}_k: k\in \{1, 2, \ldots, n\} \}}
%& = \prod_{k=1}^n p_{\boldsymbol{Y}_k-\boldsymbol{X}_k|\{Y_\ell-X_\ell\}_{\ell=1}^{k-1}} \notag\\
%& \stackrel{\eqref{distributionOfYk-Xk}}{=} \prod_{k=1}^n \phi_{(\boldsymbol{Y}_k-\boldsymbol{X}_k; 0,1)}. \label{independentNoises}
% \end{align}
\medskip
\begin{Definition} \label{defErrorProbability}
For an $(n, M, P)$-feedback code, we can calculate according to~\eqref{memorylessStatement} the \textit{average probability of decoding error} defined as $\Pr\big\{\hat W \ne W\big\}$.
We call an $(n, M, P)$-feedback code with average probability of decoding error no larger than $\varepsilon$ an $(n, M, P, \varepsilon)$-feedback code.
\end{Definition}
\medskip
Define
\[
M_{\mathrm{fb}}^*(n, \varepsilon, P) \stackrel{\text{def}}{=} \max\left\{M\in \mathbb{N} \left|\: \text{There exists an $(n, M, P, \varepsilon)$-feedback code}\right.\right\}.
\]
%\begin{Definition} \label{defAchievableRate}
%A rate $R$ is \textit{$\varepsilon$-achievable} if there exists a sequence of $(n, M, P, \varepsilon)$-feedback codes such that
%\[
%\liminf\limits_{n\rightarrow \infty}\frac{1}{n}\log M \ge R.
%\]
%%and
%%\[
%%\limsup\limits_{n\rightarrow \infty}\varepsilon \le \varepsilon.
%%\]
%\end{Definition}
\begin{Definition}\label{defCapacity}
Let $\varepsilon\in (0,1)$. The $\varepsilon$-capacity of the parallel Gaussian channel with feedback, denoted by $C_\varepsilon^{\text{fb}}$, is defined to be
\[
C_\varepsilon^{\text{fb}} \stackrel{\text{def}}{=} \liminf\limits_{n\rightarrow \infty}\frac{1}{n}\log M_{\mathrm{fb}}^*(n, \varepsilon, P).
\]
The capacity is defined to be
\[
C_0^{\text{fb}}\stackrel{\text{def}}{=} \inf_{\varepsilon > 0} C_\varepsilon^{\text{fb}}.
\]
\end{Definition}
%\medskip
%\begin{Definition} \label{defDispersionAchievable}
%A real number~$D$ is said to be second-order $\varepsilon$-achievable if there exists a sequence of $(n, M, P, \varepsilon)$-feedback codes such that
%\[
%\liminf\limits_{n\rightarrow \infty}\frac{1}{\sqrt{n}}(\log M-nC_\varepsilon^{\text{fb}}) \ge D.
%\]
%\end{Definition}

\begin{Definition} \label{defDispersion}
Let $\varepsilon\in (0,1)$.  The $\varepsilon$-second-order coding rate of the parallel Gaussian channel with feedback, denoted by $\mathrm{L}_\varepsilon^{\text{fb}}$, is defined to be
\[
\mathrm{L}_\varepsilon^{\text{fb}} \stackrel{\text{def}}{=} \liminf\limits_{n\rightarrow \infty}\frac{1}{\sqrt{n}}\left(\log M_{\mathrm{fb}}^*(n, \varepsilon, P)-nC_\varepsilon^{\text{fb}}\right).
\]
\end{Definition}

Recall how $\mathrm{C}(\mathbf{P}^*)$ and $\mathrm{V}(\mathbf{P}^*)$ are determined through \eqref{defCPparallelNFB}, \eqref{sumPell=P}, \eqref{pEllValue}, \eqref{pEllValue*} and \eqref{defVP}.
%Recall the definition of $\mathrm{C}(\mathbf{P}^*)$ in \eqref{defCPparallelNFB}.
%It is well known that $C_0=\mathrm{C}(\mathbf{P}^*)$.
Since the capacity of the parallel Gaussian channel without feedback is $\mathrm{C}(\mathbf{P}^*)$ (see, e.g., \cite{Sha49} and \cite[Sec.\ 3.4.3]{elgamal}) and an introduction of an extra noiseless feedback link does not increase the capacity~(see, e.g., \cite{Sha56} and \cite[Sec.~9.6]{CoverBook}), it follows that
\begin{equation}
C_0^{\text{fb}}=\mathrm{C}(\mathbf{P}^*). \label{wellKnownFBCapacity}
\end{equation}
%The parallel Gaussian channel with feedback is characterized by the conditional probability density function $q_{\boldsymbol{Y}|\boldsymbol{X}}$ satisfying
%
%for all $\mathbf{x}\in \mathbb{R}^L$ and $\mathbf{y}\in\mathbb{R}^L$.
% The formal definitions of the parallel Gaussian  channels with feedback and the corresponding $(n, M, P, \varepsilon)$-feedback code are similar  to Definitions~\ref{defAWGNchannel} and \ref{defErrorProbability} respectively, and hence they are omitted. We will use the following proposition concerning noise random variables extensively. The proof of the proposition can be established in a standard way using \eqref{defAWGNchannelParallel} and hence is omitted.
%
%\medskip
%\begin{Proposition} \label{propositionIndepNoises}
%Fix any $(n, M, P, \varepsilon)$-feedback code and let $p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n, \hat W}$ denote the probability distribution induced by the code. Then, the following two statements hold for each $k\in\{1, 2, \ldots, n\}$:
%\begin{enumerate}
%\item[(i):]$
%p_{\boldsymbol{X}^k, \boldsymbol{Y}^{k-1},\{Y_{\ell,k}-X_{\ell,k}\}_{\ell\in\mathcal{L}}} =  p_{\boldsymbol{X}^k, \boldsymbol{Y}^{k-1}} \prod_{\ell=1}^L p_{Y_{\ell,k}-X_{\ell,k}}$.
%\item[(ii):]  For each $\ell\in \mathcal{L}$, $\E_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}}[Y_{\ell,k}-X_{\ell,k}]=0$ and $\E_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}}[(Y_{\ell,k}-X_{\ell,k})^2]=N_\ell$.
%\end{enumerate}
%\end{Proposition}
%\medskip%
%, the capacity of the parallel Gaussian channel with feedback~\cite{Sha56}. well known and is achieved by the optimal power allocation among the~$L$ channels obtained from the water-filling algorithm \cite[Chapter 9.4]{CoverBook}, which yields $L+1$ real numbers denoted by $\Lambda$, $P_1$, $P_2$, $\ldots$, $P_L$ that satisfy
%\begin{equation}
%\sum_{\ell=1}^L P_\ell = P \label{condition(i)}
%\end{equation}
%and
%\begin{equation}
%P_\ell = \max\{0,\Lambda - N_\ell\} \label{condition(ii)}
%\end{equation}
%for each $\ell\in \mathcal{L}$.
Before stating our main result, recall that $\Phi: (-\infty, \infty)\rightarrow (0,1)$ is the cdf of the standard normal distribution. Since $\Phi$ is strictly increasing on $(-\infty, \infty)$, the inverse of $\Phi$ is well-defined and is denoted by $\Phi^{-1}$. The following theorem is the main result in this paper.
\medskip
\begin{Theorem} \label{thmMainResult}
Fix an $\varepsilon \in (0,1)$.
Then,
\begin{equation}
C_\varepsilon^{\text{fb}} = \mathrm{C}(\mathbf{P}^*)
\end{equation}
and the $\varepsilon$-second-order coding rate satisfies
\begin{equation}
\mathrm{L}_\varepsilon^{\text{fb}} \le \sqrt{\mathrm{V}(\mathbf{P}^*)}\, \Phi^{-1}(\varepsilon).
%\log M_{\mathrm{fb}}^*(n, \varepsilon, P) \le n \mathrm{C}(P) + +  \frac{1}{2}\log n + \kappa. \label{cutsetStatement}
\end{equation}
\end{Theorem}
\medskip

Combining \eqref{eqn:asymp_expans} and Theorem~\ref{thmMainResult}, we complete the characterizations of the first- and second-order asymptotics of the parallel Gaussian channel with feedback as shown in~\eqref{eqn:asymp_expans_fb}.

%In order to prove our main theorem, we need to leverage important properties of the non-asymptotic quantities in binary hypothesis testing and we also need to construct so-called {\em simulating output distributions}. These preliminaries   are contained in Section~\ref{sectionPrelim}. The details of the proof of Theorem~\ref{thmMainResult} are provided in Section~\ref{sectionMainResult}.

\section{Preliminaries for the Proof of Theorem \ref{thmMainResult}} \label{sectionPrelim}
\subsection{Binary Hypothesis Testing} \label{sectionBHT}
The following definition concerning the non-asymptotic fundamental limits of a simple binary hypothesis test is standard. See for example  \cite[Section~2.3]{Pol10}.
\medskip
\begin{Definition}\label{defBHTDivergence}
Let $p_{X}$ and $q_{X}$ be two probability distributions on some common alphabet $\mathcal{X}$. Let
\[
\mathcal{A}(\{0,1\}|\mathcal{X})\stackrel{\text{def}}{=} \{
r_{Z|X}: \text{$Z$ and $X$ assume values in $\{0,1\}$ and $\mathcal{X}$ respectively}\}
\]
be the set of randomized binary hypothesis tests between $p_{X}$ and $q_{X}$ where $\{Z=0\}$ indicates the test chooses $q_X$, and let $\delta\in [0,1]$ be a real number. The minimum type-II error in a simple binary hypothesis test between $p_{X}$ and $q_{X}$ with type-I error less than $1-\delta$ is defined as
\begin{align}
 \beta_{\delta}(p_X\|q_X) \stackrel{\text{def}}{=}
\inf\limits_{\substack{r_{Z|X} \in \mathcal{A}(\{0,1\}|\mathcal{X}): \\ \int_{\mathcal{X}}r_{Z|X}(1|x)p_X(x)\, \mathrm{d}x\ge \delta}} \int_{\mathcal{X}}r_{Z|X}(1|x)q_X(x)\, \mathrm{d}x.\label{eqDefISDivergence}
\end{align}
\end{Definition}\medskip
The existence of a minimizing test $r_{Z|X}$ is guaranteed by the Neyman-Pearson lemma.

We state in the following lemma and proposition some important properties of $\beta_{\delta}(p_X\|q_X)$, which are crucial for the proof of Theorem~\ref{thmMainResult}. The proof of the following lemma can be found  in, for example, \cite[Lemma~1]{Wang09}.
\medskip
\begin{Lemma}\label{lemmaDPI} Let $p_{X}$ and $q_{X}$ be two probability distributions on some $\mathcal{X}$, and let $g$ be a function whose domain contains $\mathcal{X}$. Then, the following two statements hold:
\begin{enumerate}
\item[1.] (Data processing inequality (DPI)) $\beta_{\delta}(p_X\|q_X) \le \beta_{\delta}(p_{g(X)}\|q_{g(X)})$.
\item[2.] For all $\xi>0$, $\beta_{\delta}(p_X\|q_X)\ge \frac{1}{\xi}\left(\delta - \int_{\mathcal{X}}p_X(x) \boldsymbol{1}\left\{ \frac{p_X(x)}{q_X(x)} \ge \xi \right\}\, \mathrm{d}x\right) $.
\end{enumerate}
\end{Lemma}
\medskip
 The proof of the following proposition can be found in~\cite[Lemma~3]{Wang09} (see also~\cite[Th.~27]{PPV10}).
 \medskip
\begin{Proposition} \label{propositionBHTLowerBound}
Let $p_{U,V}$ be a probability distribution defined on $\mathcal{W}\times \mathcal{W}$ for some finite alphabet $\mathcal{W}$, and let $p_U$ be the marginal distribution of $p_{U,V}$. In addition, let $q_{V}$ be a distribution defined on $\mathcal{W}$. Suppose $p_{U}$ is the uniform distribution, and let
\begin{equation}
\alpha = \Pr\{U\ne V\} \label{defAlpha}
\end{equation}
be a real number in $[0, 1]$ where $(U,V)$ is distributed according to $p_{U,V}$. Then,
\begin{equation}
|\mathcal{W}| \le 1/\beta_{1-\alpha}(p_{U,V}\|p_{U} q_{V}). \label{propositionBHTLowerBoundEq1}
\end{equation}
\end{Proposition}
\subsection{Modification of Power Allocation among the Parallel Channels}
For each transmitted codeword $\mathbf{x}^n\in \mathbb{R}^{L\times n}$, we can view $\sum_{k=1}^n x_{\ell,k}^2$ as the power allocated to the $\ell^{\text{th}}$ channel for each $\ell\in\mathcal{L}$. In the proof of Theorem~\ref{thmMainResult}, an early step is to discretize the power allocated to the $L$ channels. To this end, we need the following definition which defines the power allocation vector of a sequence $\mathbf{x}^n\in \mathbb{R}^{L\times n}$.
\medskip
\begin{Definition} \label{definitionPowerType}
The \emph{power allocation mapping} $\phi : \mathbb{R}^{L\times n} \rightarrow \mathbb{R}_+^L$ is defined as
\begin{equation}
\phi(\mathbf{x}^n)=\frac{1}{n}\left[\sum_{k=1}^n x_{1,k}^2\ \sum_{k=1}^n x_{2,k}^2\ \ldots \ \sum_{k=1}^n x_{L,k}^2\right]^t.
\end{equation}
We call $\phi(\mathbf{x}^n)$ the \emph{power type of $\mathbf{x}^n$}.
\end{Definition}

\medskip
The proof of Theorem~\ref{thmMainResult} involves modifying a given length-$n$ code so that useful bounds on the performance of the given code can be obtained by analyzing the modified code. More specifically, the encoding functions the given code are modified so that the power type of the random codeword generated by the modified code always falls into some small bounding box. The specific modification of the encoding functions is described in the following definition.
\medskip
\begin{Definition} \label{definitionTransformedCode}
Given an $(n, M, P)$-feedback code, let $\mathcal{W}$, $\{f_{\ell,k}|1\le \ell\le L, 1\le k\le n\}$ and $\varphi$ be the corresponding message alphabet, encoding functions and decoding function respectively. In addition, let $\gamma \ge 0$ and $\mathbf{s}=[s_1\ s_2\ \ldots \ s_L]\in \mathbb{R}_+^L$ such that $\sum_{\ell=1}^L s_\ell = P$. Then, the \emph{$(\gamma, \mathbf{s})$-modified code based on the $(n, M, P)$-feedback code} consists of the following message alphabet, encoding functions and decoding function which are denoted by $\tilde{\mathcal{W}}$, $\{\tilde f_{\ell,k}|1\le \ell\le L, 1\le k\le n\}$ and $\tilde \varphi$ respectively:
\\
\textbf{1)}  A message set $\tilde{\mathcal{W}}=\mathcal{W}$ at node~$\mathrm{s}$. Message $W$ is uniform on $\tilde{\mathcal{W}}$.
\\
\textbf{2)} An encoding function
\[
\tilde f_{\ell,k} : \mathcal{W}\times \mathbb{R}^{L\times (k-1)}\rightarrow \mathbb{R}
 \]
 for each $\ell\in\mathcal{L}$ and each $k\in\{1, 2, \ldots, n\}$, which is defined as follows. For each $w\in \mathcal{W}$ and each $\mathbf{y}^{k-1}\in  \mathbb{R}^{L\times (k-1)}$, define $\tilde f_{\ell,k}$ in a recursive manner in this order
$
 \tilde f_{1,1}, \tilde f_{2, 1}, \ldots, \tilde f_{L, 1}, \ldots,  \tilde f_{1,n}, \tilde f_{2, n}, \ldots, \tilde f_{L, n}
$
 as follows: For each $k=1, 2, \ldots, n-1$, define $\tilde f_{\ell,k}$ recursively for $\ell=1, 2, \ldots, L$ as
 \begin{align}
\tilde f_{\ell,k}(w, \mathbf{y}^{k-1})=\begin{cases}
f_{\ell,k}(w, \mathbf{y}^{k-1}) & \parbox[t]{4 in}{if $f_{\ell,k}(w, \mathbf{y}^{k-1})^2+\sum\limits_{i=1}^{k-1} \tilde f_{\ell,i}(w, \mathbf{y}^{i-1})^2 \le n(s_\ell+\gamma)$,
%\\  and
%$|f_{\ell,k}(w, \mathbf{y}^{k-1})|^2+ \sum\limits_{j=1}^{\ell-1}|\tilde f_{j,k}(w, \mathbf{y}^{k-1})|^2+\sum\limits_{j=1}^{L}\sum\limits_{i=1}^{k-1} |\tilde f_{j,i}(w, \mathbf{y}^{i-1})|^2  \le nP$,
}\\ 0 & \parbox[t]{4 in}{if $f_{\ell,k}(w, \mathbf{y}^{k-1})^2+\sum\limits_{i=1}^{k-1} \tilde f_{\ell,i}(w, \mathbf{y}^{i-1})^2 > n(s_\ell+\gamma)$.
%\\ or
%$|f_{\ell,k}(w, \mathbf{y}^{k-1})|^2+ \sum\limits_{j=1}^{\ell-1}|\tilde f_{j,k}(w, \mathbf{y}^{k-1})|^2+\sum\limits_{j=1}^{L}\sum\limits_{i=1}^{k-1} |\tilde f_{j,i}(w, \mathbf{y}^{i-1})|^2  > nP$.
}
 \end{cases} \label{defTildeFeq1}
\end{align}
It follows from~\eqref{defTildeFeq1} that
\begin{equation}
\Pr\left\{\bigcap_{\ell=1}^L\bigg\{\sum\limits_{i=1}^{n-1} \tilde f_{\ell,i}(W, \boldsymbol{Y}^{i-1})^2 \le n(s_\ell+\gamma)\bigg\}\right\}=1 \label{defTildeFeq1*}
\end{equation}
and
\begin{equation}
\Pr\left\{\sum\limits_{\ell=1}^{L}\sum\limits_{i=1}^{n-1} \tilde f_{\ell,i}(W, \boldsymbol{Y}^{i-1})^2 \le \sum\limits_{\ell=1}^{L}\sum\limits_{i=1}^{n-1} f_{\ell,i}(W, \boldsymbol{Y}^{i-1})^2\right\}=1. \label{defTildeFeq1**}
\end{equation}
 In addition, in view of~\eqref{defTildeFeq1*}, we define $\tilde f_{\ell,n}$ recursively for $\ell=1, 2, \ldots, L-1$ as follows:
 \begin{align}
&\tilde f_{\ell,n}(w, \mathbf{y}^{n-1})\notag\\*
&=\begin{cases}
f_{\ell,n}(w, \mathbf{y}^{n-1}) & \parbox[t]{5 in}{if $f_{\ell,n}(w, \mathbf{y}^{n-1})^2+\sum\limits_{i=1}^{n-1} \tilde f_{\ell,i}(w, \mathbf{y}^{i-1})^2 \in[n(s_\ell-L\gamma), n(s_\ell+\gamma)]$,}\\
 \sqrt{n(s_\ell-L\gamma) - \sum\limits_{i=1}^{n-1} \tilde f_{\ell,i}(w, \mathbf{y}^{i-1})^2}& \parbox[t]{5 in}{if $f_{\ell,n}(w, \mathbf{y}^{n-1})^2+\sum\limits_{i=1}^{n-1} \tilde f_{\ell,i}(w, \mathbf{y}^{i-1})^2 < n(s_\ell-L\gamma)$,}
 \\
\sqrt{n(s_\ell+\gamma) - \sum\limits_{i=1}^{n-1} \tilde f_{\ell,i}(w, \mathbf{y}^{i-1})^2} &   \parbox[t]{5 in}{if $f_{\ell,n}(w, \mathbf{y}^{n-1})^2+\sum\limits_{i=1}^{n-1} \tilde f_{\ell,i}(w, \mathbf{y}^{i-1})^2 > n(s_\ell-L\gamma)$.} 
 \end{cases} \label{defTildeFeq2}
\end{align}
Combining~\eqref{defTildeFeq1} and \eqref{defTildeFeq2}, we conclude that
\begin{align}
\Pr\Bigg\{\bigcap\limits_{\ell=1}^{L-1}\bigg\{\sum\limits_{i=1}^{n} \tilde f_{\ell,i}(W, \boldsymbol{Y}^{i-1})^2 \in[n(s_\ell-L\gamma), n(s_\ell+\gamma)]\bigg\}\Bigg\}=1.\label{defTildeFeq2*}
\end{align}
On the other hand, it follows from~\eqref{defTildeFeq1**}, \eqref{defTildeFeq2}, the fact $\Pr\left\{\sum_{\ell=1}^L \sum_{k=1}^n f_{\ell,i}(W, \boldsymbol{Y}^{i-1})^2  \le n P\right\}=1$ and the assumption $\sum_{\ell=1}^L s_\ell = P$ that
\begin{align}
\Pr\Bigg\{\hspace{-0.25 in}\sum\limits_{\substack{(\ell,i)\in\\\hspace{0.25 in} \mathcal{L}\times\{1, 2, \ldots, n\} \setminus\{(L, n)\}}} \hspace{-0.55 in}\tilde f_{\ell,i}(W, \boldsymbol{Y}^{i-1})^2  \le nP\Bigg\}=1. \label{defTildeFeq2**}
\end{align}
%On the other hand, it follows from ... Using~\eqref{defTildeFeq2**}, the fact that ,
Finally, in view of~\eqref{defTildeFeq2**}, we define $\tilde f_{L,n}$ as
 \begin{align}
&\tilde f_{L,n}(w, \mathbf{y}^{n-1})\notag\\*
&=\begin{cases}
f_{L,n}(w, \mathbf{y}^{n-1}) & \parbox[t]{4 in}{if $f_{L,n}(w, \mathbf{y}^{n-1})^2+\hspace{-0.25 in}\sum\limits_{\substack{(\ell,i)\in\\\hspace{0.25 in} \mathcal{L}\times\{1, 2, \ldots, n\} \setminus\{(L, n)\}}} \hspace{-0.55 in}\tilde f_{\ell,i}(w, \mathbf{y}^{i-1})^2 =nP$,}\\
 \sqrt{nP - \hspace{-0.25 in}\sum\limits_{\substack{(\ell,i)\in\\\hspace{0.25 in} \mathcal{L}\times\{1, 2, \ldots, n\} \setminus\{(L, n)\}}} \hspace{-0.55 in}\tilde f_{\ell,i}(w, \mathbf{y}^{i-1})^2}& \parbox[t]{4 in}{if $f_{L,n}(w, \mathbf{y}^{n-1})^2+\hspace{-0.25 in}\sum\limits_{\substack{(\ell,i)\in\\\hspace{0.25 in} \mathcal{L}\times\{1, 2, \ldots, n\} \setminus\{(L, n)\}}} \hspace{-0.55 in}\tilde f_{\ell,i}(w, \mathbf{y}^{i-1})^2 < nP$.}
 \end{cases} \label{defTildeFeq3}
\end{align}
Combining~\eqref{defTildeFeq2*}, \eqref{defTildeFeq3} and the assumption that $\sum_{\ell=1}^L s_\ell = P$, we have
\begin{align}
\Pr\Bigg\{\bigcap\limits_{\ell=1}^{L}\bigg\{\sum\limits_{i=1}^{n} \tilde f_{\ell,i}(W, \boldsymbol{Y}^{i-1})^2 \in [n(s_\ell-L\gamma), n(s_\ell+L^2\gamma)]\bigg\}\cap\Bigg\{\sum\limits_{\ell=1}^{L}\sum\limits_{i=1}^{n}\tilde f_{\ell,i}(W, \boldsymbol{Y}^{i-1})^2  = nP\Bigg\}\Bigg\}=1. \label{defTildeFeq3*}
\end{align}
%and the peak power constraint~\eqref{peakPowerConstraint} holds.
\textbf{3)} A decoding function
%\[
%\tilde \varphi :
%\mathbb{R}^{L\times n} \rightarrow \mathcal{W},
% \]
%where
\[
\tilde \varphi = \varphi
\]
for providing an estimate of $W$ at node~$\mathrm{d}$. \hfill $\blacksquare$
\end{Definition}
\medskip
\begin{Remark}
The basic idea behind transforming a code in Definition~\ref{definitionTransformedCode} is simple. Suppose we are given an $(n, M, P)$-feedback code, a $\gamma \ge 0$ and an $\mathbf{s}=[s_1\ s_2\ \ldots \ s_L]\in \mathbb{R}_+^L$ such that $\sum_{\ell=1}^L s_\ell = P$. Then, the $(\gamma, \mathbf{s})$-modified code is formed by
\begin{enumerate}
\item[(i)] truncating a transmitted codeword if the power transmitted over the $\ell^{\text{th}}$ channel exceeds $n(s_\ell + \gamma)$, which can be seen from~\eqref{defTildeFeq1} and the third clause of~\eqref{defTildeFeq2};
     \item[(ii)] boosting the power of the transmitted codeword if the power transmitted over the $\ell^{\text{th}}$ channel falls below $n(s_\ell -L \gamma)$, which can be seen from the second clause of~\eqref{defTildeFeq2};
         \item[(iii)] adjusting the last symbol transmitted over the $L^{\text{th}}$ channel (i.e., $X_{L, n}$) so that the total transmitted power is exactly equal to~$nP$, which can be seen from the second clause of~\eqref{defTildeFeq3}.
\end{enumerate}
\end{Remark}
\medskip

Given an $(n, M, P)$-feedback code, we consider the corresponding $(\gamma, \mathbf{s})$-modified code constructed in Definition~\ref{definitionTransformedCode} and let $\tilde p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n, \hat W}$ be the distribution induced by the modified code. By~\eqref{defTildeFeq3*}, we see that
\begin{align}
\Pr_{\tilde p_{\boldsymbol{X}^n}}\left\{\bigcap\limits_{\ell=1}^{L}\bigg\{\sum\limits_{k=1}^{n} X_{\ell, k}^2 \in[n(s_\ell-L^2\gamma), n(s_\ell+L\gamma)]\bigg\}\cap\bigg\{\sum\limits_{\ell=1}^{L}\sum\limits_{k=1}^{n}X_{\ell, k}^2  = nP\bigg\}\right\}=1. \label{modifiedCodeProperty*}
\end{align}
Define the $\Delta$-bounding box
\begin{equation}
\Gamma^{(\Delta)}(\mathbf{s})\stackrel{\text{def}}{=} [s_1-\Delta,s_1+\Delta]\times [s_2-\Delta,s_2+\Delta]\times \ldots \times [s_L-\Delta,s_L+\Delta]. \label{defGammaS}
\end{equation}
for each $\gamma\ge 0$ and each $\mathbf{s}\in \mathbb{R}_+^L$.
It then follows from~\eqref{modifiedCodeProperty*} that
\begin{align}
\Pr_{\tilde p_{\boldsymbol{X}^n}}\left\{\left\{\phi(\boldsymbol{X}^n)\in\Gamma^{(L^2\gamma)}(\mathbf{s})\right\}\cap\bigg\{\sum\limits_{\ell=1}^{L}\sum\limits_{k=1}^{n}X_{\ell, k}^2  = nP\bigg\}\right\}=1. \label{modifiedCodeProperty}
\end{align}
The following lemma is a natural consequence of Definition~\ref{definitionTransformedCode}, and the proof is deferred to Appendix~\ref{appendixA}.
\medskip
\begin{Lemma}\label{lemmaTransformedCode}
Given an $(n, M, P)$-feedback code, let $p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}$ be the distribution induced by the code. Fix any $\gamma\ge 0$ and any $\mathbf{s}\in \mathbb{R}_+^L$ such that $\sum_{\ell=1}^L s_\ell = P$, and let $\tilde p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}$ be the distribution induced by the $(\gamma, \mathbf{s})$-modified code based on the $(n, M, P)$-feedback code. Then, we have
\begin{align}
\int_{\mathcal{A}}p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}(\mathbf{x}^n, \mathbf{y}^n)\mathbf{1}\left\{\phi(\mathbf{x}^n)\in \Gamma^{(\gamma)}(\mathbf{s})\right\} \mathbf{1}\left\{\sum_{\ell=1}^L\sum_{k=1}^n x_{\ell,k}^2=nP\right\}\mathrm{d}\mathrm{x}^n\mathrm{d}\mathrm{y}^n \le \int_{\mathcal{A}} \tilde p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}(\mathbf{x}^n, \mathbf{y}^n)\mathrm{d}\mathrm{x}^n\mathrm{d}\mathrm{y}^n \label{lemmaTransformedCodeSt}
\end{align}
for all Borel measurable~$\mathcal{A}\subseteq\mathbb{R}^{L\times n}\times \mathbb{R}^{L\times n}$.
\end{Lemma}

\subsection{Curtiss' Theorem} \label{sectionCurtiss}
Curtiss' theorem~\cite[Th.~3]{curtiss} states that convergence of moment generating functions leads to convergence in distribution. The formal statement is reproduced below.
\smallskip
 \begin{Theorem}[Curtiss' theorem] \label{thmCurtiss}
 Let $U^{(n)}$ be a sequence of real-valued random variables. Suppose there exists a random variable $V$ such that
 \begin{align}
\lim_{n\rightarrow\infty} \E\left[e^{tU^{(n)}}\right] = \E\left[e^{t V}\right] \label{st1ThmCurtiss}
 \end{align}
 for all $t\in\mathbb{R}$. Then,
 \begin{align}
\lim_{n\rightarrow\infty} \Pr\{U^{(n)}\le a\} =  \Pr\{V \le a\}
 \end{align}
 for every $a\in\mathbb{R}$ at which $a \mapsto \Pr\{V \le a\}$ is continuous.
 \end{Theorem}
\smallskip

 In contrast to the more well-known L\'evy's continuity theorem~\cite[Sec.~18.1]{Williams1991}, \eqref{st1ThmCurtiss} of Theorem~\ref{thmCurtiss} is required to be true for all real rather than purely imaginary~$t$.



% the  distribution of the sum of random variables in the information spectrum term converges to some distribution generated from a sum of i.i.d.\ random variables (i.e., \eqref{levyThmEq1}), thus facilitating the use of the usual central limit theorem~\cite{feller}. The details can be found in Section~\ref{stepBInProof}.
%The proof of Theorem~\ref{thmMainResult} relies on the following lemma, which provides an upper bound on the second-order term of the asymptotic expansion of $\log M_{\mathrm{fb}}^*(n, \varepsilon, P)$.
%%we let
%%\begin{equation}
%%\mathrm{C}(\mathbf{P}^*) \stackrel{\text{def}}{=} \sum_{\ell=1}^L \mathrm{C}(P_\ell/N_\ell) \qquad\mbox{bits per channel use} \label{defCPparallelNFB}
%%\end{equation}
%%%and
%%%\begin{equation}
%%%\mathrm{V}(\mathbf{P}^*) \stackrel{\text{def}}{=} \sum_{\ell=1}^L \mathrm{V}(P_\ell/N_\ell) \qquad\mbox{bits$^2$ per channel use}, \label{defVPparallel}
%%%\end{equation}
%%be the capacity of the parallel Gaussian channel \cite[Chapter 9.4]{CoverBook}.
%\medskip
%\begin{Lemma} \label{lemmaMainResult}
%Fix an $\varepsilon \in (0,1)$ and an $(n, M_{\mathrm{fb}}^*(n, \varepsilon, P), P, \varepsilon)$-code, and let $p_{W,\boldsymbol{X}^n, \boldsymbol{Y}^n, \hat W}$ be the distribution induced by the code. For each $\ell\in\mathcal{L}$, let
%\begin{equation}
%S_\ell \stackrel{\text{def}}{=} \E\left[\frac{1}{n}\sum_{k=1}^n X_{\ell, k}^2\right]
%\end{equation}
%and
%define
%\begin{equation}
%\mathbf{S}\stackrel{\text{def}}{=}[S_1\ S_2\ \ldots\ S_L]^t.
%\end{equation}
%% and
%%Recall that the values  $\Lambda$, $P_1, P_2, \ldots, P_L$ and $\mathrm{C}(\mathbf{P}^*)$ are  determined by \eqref{condition(i)}, \eqref{condition(ii)} and \eqref{defCPparallelNFB}.
%There exists a constant $\kappa$ not depending on~$n$ such that for each sufficiently large $n\in \mathbb{N}$,
%\begin{equation}
%\log M_{\mathrm{fb}}^*(n, \varepsilon, P) \le n \mathrm{C}(\mathbf{S}) + \kappa\sqrt{n}. \label{cutsetStatementParallel}
%\end{equation}
%\end{Lemma}
%\subsection{Simulating Output Distribution} \label{sectionSimulatingDistribution}
%Proposition~\ref{propositionBHTLowerBound} and Statement~2 of Lemma~\ref{lemmaDPI} together imply a lower bound for the error probability, and the lower bound holds for all $q_V$. Therefore, we are motivated to choose a \textit{simulating output distribution} $q_V$ which is almost the same as the output distribution chosen in \cite[Section 4.2.2]{Pol10} so that the right hand side of \eqref{propositionBHTLowerBoundEq1} can be simplified. The construction of the simulating output distribution is contained in the following lemma.
%
%\begin{Lemma} \label{lemmaSimulatingDistribution}
% Given an $(n, M, P, \varepsilon)$-feedback code for the AWGN channel, let $p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n, \hat W}$ be the probability distribution induced by the code according to \eqref{memorylessStatement}. Then there exists a probability distribution $s_{Y^n, \hat W}$ that satisfies the following properties:
%  \begin{enumerate}
% %\item[(i)] $s_{W} = p_{W}$
% \item[(i)] $s_{\hat W|Y^n} = p_{\hat W|Y^n}$
% \item [(ii)] $s_{Y^n} = \prod_{k=1}^n s_{\boldsymbol{Y}_k}$
%     \item[(iii)] For each $k\in\{1, 2, \ldots, n\}$, $s_{\boldsymbol{Y}_k}(y_k) = \phi_{0, 1+P}(y_k)$ for all $y_k\in \mathbb{R}$.
% \end{enumerate}
%We call $s_{Y^n, \hat W}$ a \textit{simulating output distribution of $p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n, \hat W}$} because $s_{Y^n, \hat W}$ captures all the important properties of $(Y^n, \hat W)$ when $(W, X^n, Y^n, \hat W)$ is generated according to the given probability distribution $p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n, \hat W}$.
%\end{Lemma}
%\begin{IEEEproof}
%Define $s_{Y^n, \hat W}$ as
%  \begin{equation}
% s_{Y^n, \hat W}(y^n, \hat w)= \left( \prod_{k=1}^n \phi_{0, 1+P}(y_k) \right) p_{\hat W|Y^n}(\hat w|y^n) \label{marginal2ndWay}
% \end{equation}
% for all $\hat W \in \mathcal{W}$ and $y^n\in \mathbb{R}^n$.
%  In order to prove Property~(i), we marginalize \eqref{marginal2ndWay} and obtain
%   \begin{equation}
% s_{Y^n}(y^n)= \prod_{k=1}^n \phi_{0, 1+P}(y_k)  \label{marginal3rdWay}
% \end{equation}
% for all $y^n\in \mathbb{R}^n$. Property~(i) then follows from \eqref{marginal2ndWay} and \eqref{marginal3rdWay}. Property (iii) follows from marginalizing \eqref{marginal3rdWay}. Property (ii) follows from \eqref{marginal3rdWay} and  Property (iii).
%\end{IEEEproof}


\section{Proof of Theorem~\ref{thmMainResult}} \label{sectionMainResult}
Fix an $\varepsilon\in (0,1)$ and choose an arbitrary sequence of $(\bar n, M_{\mathrm{fb}}^*(\bar n, \varepsilon, P), P, \varepsilon)$-feedback codes. Since
\begin{equation}
C_\varepsilon^{\text{fb}} \ge \mathrm{C}(\mathbf{P}^*)
\end{equation}
by~\eqref{wellKnownFBCapacity}, it suffices to show that
\begin{equation}
\liminf\limits_{n\rightarrow \infty}\frac{1}{\sqrt{n}}\big(\log M_{\mathrm{fb}}^*(n, \varepsilon, P)-n\mathrm{C}(\mathbf{P}^*)\big) \le \sqrt{\mathrm{V}(\mathbf{P}^*)}\, \Phi^{-1}(\varepsilon+ \tau)  \label{goalInProofMainResult}
\end{equation}
for all $\tau>0$. To this end, fix an arbitrary $\tau>0$.
\subsection{Discretizing the Power Allocation Vectors by Appending Symbols} \label{stepAinProof}
 Using Definition~\ref{defCode}, we have
\begin{equation*}
\Pr\left\{\sum_{\ell=1}^L\sum_{k=1}^{\bar n}X_{\ell,k}^2 \le \bar nP\right\} = 1 %\label{powerConstraintInProof}
\end{equation*}
for the chosen $(\bar n, M_{\mathrm{fb}}^*(\bar n, \varepsilon, P), P, \varepsilon)$-feedback code for each~$\bar n\in \mathbb{N}$.
  Given the chosen $(\bar n, M_{\mathrm{fb}}^*(\bar n, \varepsilon, P), P, \varepsilon)$-feedback code, we can always construct an $(\bar n+L, M_{\mathrm{fb}}^*(\bar n, \varepsilon, P), P, \varepsilon)$-feedback code by appending a carefully chosen tuple $(\boldsymbol{X}_{\bar n+1}, \boldsymbol{X}_{\bar n+2}, \ldots, \boldsymbol{X}_{\bar n+L})$ to each transmitted codeword $\boldsymbol{X}^{\bar n}$ generated by the $(\bar n, M_{\mathrm{fb}}^*(\bar n, \varepsilon, P), P, \varepsilon)$-feedback code such that
\begin{equation}
\Pr\left\{\sum_{k=1}^{\bar n+L}X_{\ell,k}^2 = P\left\lceil\frac{1}{P}\sum_{k=1}^{\bar n}X_{\ell,k}^2\right\rceil  \text{ for all $\ell\in \mathcal{L}$}\right\} = 1, %\label{powerConstraintInProof}
\end{equation}
which implies that
\begin{equation}
\Pr\left\{\sum_{k=1}^{\bar n+L}X_{\ell,k}^2 \text{ is a multiple of~$P$ for all $\ell\in \mathcal{L}$ and }\sum_{\ell=1}^L\sum_{k=1}^{\bar n+L}X_{\ell,k}^2 \le (\bar n+L)P\right\} = 1. \label{powerConstraintInProof**}
\end{equation}
In addition, given the $(\bar n+L, M_{\mathrm{fb}}^*(\bar n, \varepsilon, P), P, \varepsilon)$-feedback code, we can always construct an $(\bar n+L+1, M_{\mathrm{fb}}^*(\bar n, \varepsilon, P), \linebreak P, \varepsilon)$-feedback code by appending a carefully chosen $\boldsymbol{X}_{\bar n+L+1}$ to each transmitted codeword $\boldsymbol{X}^{\bar n+L}$ generated by the $(\bar n +L, M_{\mathrm{fb}}^*(\bar n, \varepsilon, P), P, \varepsilon)$-feedback code such that
\begin{equation}
\Pr\left\{\sum_{k=1}^{\bar n+L+1}X_{\ell,k}^2 \text{ is a multiple of~$P$ for all $\ell\in \mathcal{L}$ and }\sum_{\ell=1}^L\sum_{k=1}^{\bar n+L+1}X_{\ell,k}^2 = (\bar n+L+1)P\right\} = 1. \label{powerConstraintInProof***}
\end{equation}
%The technique of transforming the peak power inequality constraint \eqref{powerConstraintInProof***} to the power equality constraint \eqref{powerConstraintInProof**} by appending an extra symbol has been employed in \cite[Lemma~39]{PPV10} and \cite[Theorem~4.4]{TanBook} (and is called the Yaglom map trick).
To simplify notation, we let
\begin{equation}
n\stackrel{\text{def}}{=}\bar n+L+1. \label{defBarN}
\end{equation}
Construct the set of power allocation vectors
 \begin{align}
 \mathcal{S}^{(n)} \stackrel{\text{def}}{=} \left\{\frac{P}{n}\cdot a^L  \left|  a^L\in \mathbb{Z}_+^L, \:\sum_{\ell=1}^L a_\ell = n \right.\right\}, \label{defSetS}
 \end{align}
 which can be viewed as a set of quantized power allocation vectors $\mathbf{s}$ with quantization level $P/n$ that satisfy the equality power constraint
 \begin{equation*}
 \sum_{\ell=1}^L s_\ell = P.
 \end{equation*}
It follows from~\eqref{defSetS}, \eqref{powerConstraintInProof***} and Definition~\ref{definitionPowerType} that
\begin{equation}
|\mathcal{S}^{(n)}| \le n^L \label{cardinalityBound}
\end{equation}
and
\begin{align}
\Pr\left\{\phi(\boldsymbol{X}^n)\in  \mathcal{S}^{(n)}\right\} = 1. \label{powerConstraintInProof****}
\end{align}

\subsection{Obtaining a Lower Bound on the Error Probability in Terms of the Type-II Error of a Hypothesis Test}
 Let $p_{W,\boldsymbol{X}^n, \boldsymbol{Y}^n, \hat W}$ be the probability distribution induced by the $(n, M_{\mathrm{fb}}^*(\bar n, \varepsilon, P), P, \varepsilon)$-feedback code constructed above for each $n\in\{L+2, L+3, \ldots\}$, where $p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n, \hat W}$ is obtained according to \eqref{memorylessStatement}.
% In view of \eqref{powerConstraintInProofnBar}, we assume without loss of generality that
% \begin{equation}
%p_{W,X^n, Y^n}(w,x^n, y^n) = p_{W,X^n, Y^n}(w,x^n, y^n)\mathbf{1}\left(\left\{\sum_{k=1}^{n}x_k^2 = nP\right\}\right) \label{powerConstraintCharacteristicFunction}
% \end{equation}
% for all $w\in\mathcal{W}$, $x^n\in \mathbb{R}^n$ and $y^n\in\mathbb{R}^n$.
  Fix an $n\in\{L+2, L+3, \ldots\} $ and the corresponding $(n, M_{\mathrm{fb}}^*(\bar n, \varepsilon, P), P, \varepsilon)$-feedback code.
Recall the definition of $P_\ell$ for each $\ell\in\mathcal{L}$ in~\eqref{pEllValue} and define the distribution
\begin{align}
r_{\boldsymbol{Y}^n, \hat W}\stackrel{\text{def}}{=} p_{\hat W|\boldsymbol{Y}^n} r_{\boldsymbol{Y}^n} \label{defDistS}
\end{align}
where\footnote{We note that even if we exclude the set of power types in the set $\Pi^{(n)}$ which is defined later in~\eqref{defSetPiN}, this leads to another valid definition of $r_{\boldsymbol{Y}^n}(\mathbf{y}^n)$.}
%Another valid choice of $r_{\boldsymbol{Y}^n}(\mathbf{y}^n)$ may exclude averaging over the power types inside the set $\Pi^{(n)}$ defined later in~\eqref{defSetPiN}.
\begin{align}
  r_{\boldsymbol{Y}^n}(\mathbf{y}^n)  \stackrel{\text{def}}{=} \frac{1}{2|\mathcal{S}^{(n)}|}\sum_{\mathbf{s}\in \mathcal{S}^{(n)}}\prod_{\ell=1}^L \prod_{k=1}^n \mathcal{N}(y_{\ell,k}; 0, s_\ell+N_\ell) + \frac{1}{2}\prod_{\ell=1}^L\prod_{k=1}^n \mathcal{N}(y_{\ell,k}; 0, P_\ell+N_\ell). \label{defDistSyk}
  \end{align}
  The choice of $r_{\boldsymbol{Y}^n}$ in~\eqref{defDistSyk} is motivated by the choice of the auxiliary output distribution in~\cite[Sec.~X-A]{Hayashi09} where DMCs are considered.
%Let $s_{Y^n, \hat W}$ be a simulating output distribution of $p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n, \hat W}$ such that $s_{Y^n, \hat W}$ satisfies all the properties in Lemma~\ref{lemmaSimulatingDistribution}.
Then, it follows from Proposition~\ref{propositionBHTLowerBound} and Definition~\ref{defCode} with the identifications $U\equiv W$, $V\equiv \hat W$, $p_{U,V}\equiv p_{W,\hat W}$, $q_V\equiv r_{\hat W}$, $|\mathcal{W}|\equiv M_{\mathrm{fb}}^*(\bar n, \varepsilon, P)$ and $\alpha\equiv \Pr\{\hat W \ne W\} \le \varepsilon$ that
 \begin{align}
\beta_{1-\varepsilon}(p_{W,\hat W}\|p_W r_{\hat W}) \le \beta_{1-\alpha}(p_{W,\hat W}\|p_W r_{\hat W}) \le 1/M_{\mathrm{fb}}^*(\bar n, \varepsilon, P). \label{eqnBHTReverseChain}
 \end{align}
  \subsection{Obtaining a Non-Asymptotic Bound from Simplifying the Type-II Error of the Binary Hypothesis Test}
Using the DPI of $\beta_{1-\varepsilon}$ by introducing $\boldsymbol{X}^n$ and $\boldsymbol{Y}^n$, we have
\begin{align}
\beta_{1-\varepsilon}(p_{W,\hat W}\|p_W r_{\hat W}) \ge \beta_{1-\varepsilon}\bigg(p_{W,\boldsymbol{X}^n,\boldsymbol{Y}^n,\hat W}\bigg\|p_W r_{\boldsymbol{Y}^n,\hat W}\prod_{k=1}^n p_{\boldsymbol{X}_k|W,\boldsymbol{Y}^{k-1}}\bigg) \label{eqnBHTFirstChain}
\end{align}
%
% \begin{align}
% & \beta_{1-\varepsilon}(p_{W,\hat W}\|p_W r_{\hat W}) \notag\\
%& =\beta_{1-\varepsilon}(p_W p_{\hat W|W}\|p_W r_{\hat W}) \notag\\
%&\stackrel{\text{(a)}}{\ge} \beta_{1-\varepsilon}(p_Wp_{\hat W, Y^n|W}\|p_W r_{\hat W, Y^n}) \notag\\
%& = \beta_{1-\varepsilon}(p_Wp_{Y^n|W}p_{\hat W|Y^n, W}\|p_Ws_{Y^n} r_{\hat W|Y^n}) \notag\\
%& \stackrel{\text{(b)}}{=} \beta_{1-\varepsilon}(p_W p_{Y^n|W}p_{\hat W|Y^n, W}\|p_W s_{Y^n} p_{\hat W|Y^n}) \notag\\
%& \stackrel{\text{(c)}}{=}\beta_{1-\varepsilon}(p_W p_{Y^n|W}p_{\hat W|Y^n}\|p_W s_{Y^n} p_{\hat W|Y^n}) \notag\\
%&\stackrel{\text{(d)}}{\ge} \beta_{1-\varepsilon}\left(p_W p_{\hat W|Y^n}p_{X^n,Y^n|W}\left\|p_Wp_{\hat W|Y^n} s_{Y^n}\prod_{k=1}^n p_{\boldsymbol{X}_k|\boldsymbol{Y}^{k-1}, W}\right.\right)  \label{eqnBHTFirstChain}
%\end{align}
%where
% \begin{enumerate}
% \item[(a)] follows from the DPI of $\beta_{1-\varepsilon}$ by introducing the channel output $Y^n$.
% \item[(b)] follows from Property~(i) in Lemma~\ref{lemmaSimulatingDistribution}.
% \item[(c)] follows from the fact that
% \[
% W  \rightarrow Y^n \rightarrow \hat W
% \]
% forms a Markov chain for the $(n, M_{\mathrm{fb}}^*(n-1, \varepsilon, P), P, \varepsilon)$-feedback code (cf.\ Definition~\ref{defCode}).
% \item[(d)] follows from the DPI of $\beta_{1-\varepsilon}$ by introducing the channel input $X^n$.
% \end{enumerate}
%  \subsection{Obtaining a Non-Asymptotic Bound from the Binary Hypothesis Testing}
where
 \begin{align}
   p_{W, \boldsymbol{X}^n , \boldsymbol{Y}^n, \hat W} =p_W p_{\hat W|\boldsymbol{Y}^n}\prod_{k=1}^n (p_{\boldsymbol{X}_k|W, \boldsymbol{Y}^{k-1}}p_{\boldsymbol{Y}_k|\boldsymbol{X}_k}) \label{eqnBHTFirstChain*}
 \end{align}
by~\eqref{memorylessStatement}.
%
% which implies that
%\begin{align}
%  p_{X^n , Y^n|W} =\prod_{k=1}^n (p_{\boldsymbol{X}_k|\boldsymbol{Y}^{k-1}, W}p_{\boldsymbol{Y}_k|\boldsymbol{X}_k}).  \label{eqnBHTFirstChain*}
% \end{align}
 Combining \eqref{eqnBHTFirstChain}, \eqref{eqnBHTFirstChain*} and~\eqref{defDistS}, we have
 \begin{align}
&  \beta_{1-\varepsilon}(p_{W,\hat W}\|p_W r_{\hat W}) \notag\\
& \ge  \beta_{1-\varepsilon}\left(p_W p_{\hat W|Y^n}\prod_{k=1}^n (p_{\boldsymbol{X}_k|\boldsymbol{Y}^{k-1}, W}p_{\boldsymbol{Y}_k|\boldsymbol{X}_k})\left\|p_W r_{\boldsymbol{Y}^n} p_{\hat W|Y^n} \prod_{k=1}^n p_{\boldsymbol{X}_k|W,\boldsymbol{Y}^{k-1}} \right.\right). \label{eqnBHT5thChain}
 \end{align}
Fix any constant $\xi_n>0$ to be specified later. Using Lemma~\ref{lemmaDPI}, \eqref{eqnBHT5thChain} and~\eqref{defChannelInDefinition*},
we have
\begin{align}
 \beta_{1-\varepsilon}(p_{W,\hat W}\|p_W r_{\hat W})  \ge \frac{1}{\xi_n}\left(1-\varepsilon - \Pr_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}}\left\{\frac{\prod_{k=1}^n q_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{Y}_k|\boldsymbol{X}_{k})}{r_{\boldsymbol{Y}^n}(\boldsymbol{Y}^n)} \ge \xi_n \right\}\right), \label{eqnBHTThirdChain}
\end{align}
which together with \eqref{eqnBHTReverseChain} implies that
\begin{align}
\log M_{\mathrm{fb}}^*(\bar n, \varepsilon, P)
 \le \log\xi_n - \log\left(1-\varepsilon -\Pr_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}}\left\{\log\bigg(\frac{\prod_{k=1}^n q_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{Y}_k|\boldsymbol{X}_{k})}{r_{\boldsymbol{Y}^n}(\boldsymbol{Y}^n)}\bigg) \ge \log \xi_n \right\}\right).\label{eqnBHTSecondChain}
\end{align}
\subsection{Splitting the Probability Term into Multiple Terms Corresponding to Different Power Types of $\boldsymbol{X}^n$}
Define\footnote{The conclusion of this proof remains unchanged if the $n^{1/6}$ term in~\eqref{defSetPiN} is replaced by $n^a$ for any $a\in(0, 1/2)$.}
\begin{align}
\Pi^{(n)} \stackrel{\text{def}}{=} \left\{\mathbf{s}\in\mathcal{S}^{(n)}  \left|  \|\mathbf{s}-\mathbf{P}^*\|_2\le \frac{1}{n^{1/6}} \right.\right\} \label{defSetPiN}
\end{align}
to be the set of power allocation vectors in $\mathcal{S}^{(n)}$ that are close to the optimal power allocation vector $\mathbf{P}^*$ (cf.\ \eqref{pEllValue*}).
Following~\eqref{eqnBHTSecondChain}, we use~\eqref{powerConstraintInProof****} to obtain
\begin{align}
&\Pr_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}}\left\{\log\bigg(\frac{\prod_{k=1}^n q_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{Y}_k|\boldsymbol{X}_{k})}{r_{\boldsymbol{Y}^n}(\boldsymbol{Y}^n)}\bigg) \ge \log \xi_n \right\}\notag\\
& = \Pr_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}}\left\{\left\{\log\bigg(\frac{\prod_{k=1}^n q_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{Y}_k|\boldsymbol{X}_{k})}{r_{\boldsymbol{Y}^n}(\boldsymbol{Y}^n)}\bigg) \ge \log \xi_n\right\}\cap\left\{\phi(\boldsymbol{X}^n)\in \Pi^{(n)} \right\} \right\} \notag\\
&\qquad + \sum_{\mathbf{s}\in \mathcal{S}^{(n)}\setminus\Pi^{(n)}}\Pr_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}}\left\{\left\{\log\bigg(\frac{\prod_{k=1}^n q_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{Y}_k|\boldsymbol{X}_{k})}{r_{\boldsymbol{Y}^n}(\boldsymbol{Y}^n)}\bigg) \ge \log \xi_n\right\}\cap\left\{\phi(\boldsymbol{X}^n)=\mathbf{s} \right\} \right\}. \label{eqnBHT3rdChain}
\end{align}
In order to bound the first term in~\eqref{eqnBHT3rdChain}, we let
\begin{equation}
\gamma\stackrel{\text{def}}{=}\frac{1}{n^{1/6}} \label{defgammaInProof}
 \end{equation}
 and define $p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{*}$ be the distribution induced by the $(\gamma, \mathbf{P}^*)$-modified code based on the $(n, M_{\mathrm{fb}}^*(\bar n, \varepsilon, P), P, \varepsilon)$-feedback code defined in Definition~\ref{definitionTransformedCode}.
% , which satisfies
% \begin{equation}
% \Pr_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{*}}\left\{\left\{\phi(\boldsymbol{X}^n)\in\Gamma^{(L^2\gamma)}(\mathbf{s})\right\}\cap\bigg\{\sum\limits_{\ell=1}^{L}\sum\limits_{k=1}^{n}X_{\ell, k}^2  = nP\bigg\}\right\}=1 \label{defDistP*}
% \end{equation}
% by~\eqref{modifiedCodeProperty}.
 Then, consider the following chain of inequalities:
\begin{align}
&\Pr_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}}\left\{\left\{\log\bigg(\frac{\prod_{k=1}^n q_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{Y}_k|\boldsymbol{X}_{k})}{r_{\boldsymbol{Y}^n}(\boldsymbol{Y}^n)}\bigg) \ge \log \xi_n\right\}\cap\left\{\phi(\boldsymbol{X}^n)\in \Pi^{(n)} \right\}\right\}\notag\\*
&\le \Pr_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{*}}\left\{\log\bigg(\frac{\prod_{k=1}^n q_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{Y}_k|\boldsymbol{X}_{k})}{r_{\boldsymbol{Y}^n}(\boldsymbol{Y}^n)}\bigg) \ge \log \xi_n\right\} \label{eqnBHT3rd1Chain(a)} \\*
& \le \Pr_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{*}}\left\{\sum_{k=1}^n\log\bigg(\frac{ q_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{Y}_k|\boldsymbol{X}_{k})}{\prod_{\ell=1}^L\mathcal{N}(Y_{\ell,k};0, P_\ell+N_\ell)}\bigg) \ge \log \xi_n - \log 2\right\} \label{eqnBHT3rd1Chain(b)}
% \\
%& = \Pr_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{*}}\left\{\sum_{k=1}^n\sum_{\ell=1}^L\log\bigg(\frac{ \mathcal{N}(Y_{\ell,k}-X_{\ell,k};0, N_\ell)}{\mathcal{N}(Y_{\ell,k};0, P_\ell+N_\ell)}\bigg) \ge \log \xi_n - \log 2\right\} \label{eqnBHT3rd1Chain(c)}
\end{align}
where
\begin{itemize}
\item \eqref{eqnBHT3rd1Chain(a)} is due to Lemma~\ref{lemmaTransformedCode} and the fact that $\Pr_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}}\left\{\sum_{\ell=1}^L\sum_{k=1}^n X_{\ell,k}^2=nP\right\}=1$ (cf.\ \eqref{defSetS} and~\eqref{powerConstraintInProof****}).
\item \eqref{eqnBHT3rd1Chain(b)} is due to the definition of $r_{\boldsymbol{Y}^n}$ in~\eqref{defDistSyk}.
%\item[$\bullet$] \eqref{eqnBHT3rd1Chain(c)} is due to the definition of $q_{\boldsymbol{Y}|\boldsymbol{X}}$ in~\eqref{defAWGNchannelParallel}.
\end{itemize}
Similarly, in order to bound the second term in~\eqref{eqnBHT3rdChain}, we let $p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{(\mathbf{s})}$ be the distribution induced by the $(0, \mathbf{s})$-modified code and consider the following chain of inequalities for each $\mathbf{s}\in \mathcal{S}^{(n)}\setminus\Pi^{(n)}$:
\begin{align}
&\Pr_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}}\left\{\left\{\log\bigg(\frac{\prod_{k=1}^n q_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{Y}_k|\boldsymbol{X}_{k})}{r_{\boldsymbol{Y}^n}(\boldsymbol{Y}^n)}\bigg) \ge \log \xi_n\right\}\cap\left\{\phi(\boldsymbol{X}^n)=\mathbf{s} \right\}\right\}\notag\\
&\le \Pr_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{(\mathbf{s})}}\left\{\log\bigg(\frac{\prod_{k=1}^n q_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{Y}_k|\boldsymbol{X}_{k})}{r_{\boldsymbol{Y}^n}(\boldsymbol{Y}^n)}\bigg) \ge \log \xi_n\right\} \label{eqnBHT4thChain(a)} \\
& \le \Pr_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{(\mathbf{s})}}\left\{\sum_{k=1}^n\log\bigg(\frac{ q_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{Y}_k|\boldsymbol{X}_{k})}{\prod_{\ell=1}^L\mathcal{N}(Y_{\ell,k};0, s_\ell+N_\ell)}\bigg) \ge \log \xi_n - \log\big(2|\mathcal{S}^{(n)}|\big)\right\} \label{eqnBHT4thChain(b)}
%& = \Pr_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{(\mathbf{s})}}\left\{\sum_{k=1}^n\sum_{\ell=1}^L\log\bigg(\frac{ \mathcal{N}(Y_{\ell,k}-X_{\ell,k};0, N_\ell)}{\mathcal{N}(Y_{\ell,k};0, s_\ell+N_\ell)}\bigg) \ge \log \xi_n - \log\big(2|\mathcal{S}^{(n)}|\big)\right\} \label{eqnBHT4thChain(c)}
\end{align}
where
\begin{itemize}
\item \eqref{eqnBHT4thChain(a)} is due to Lemma~\ref{lemmaTransformedCode}.
\item \eqref{eqnBHT4thChain(b)} is due to the definition of $r_{\boldsymbol{Y}^n}$ in~\eqref{defDistSyk}.
%\item[$\bullet$] \eqref{eqnBHT4thChain(c)} is due to the definition of $q_{\boldsymbol{Y}|\boldsymbol{X}}$ in~\eqref{defAWGNchannelParallel}.
\end{itemize}
Combining~\eqref{eqnBHT3rdChain}, \eqref{eqnBHT3rd1Chain(b)}, \eqref{eqnBHT4thChain(b)} and the definition of $q_{\boldsymbol{Y}|\boldsymbol{X}}$ in~\eqref{defAWGNchannelParallel} followed by letting
\begin{equation}
Z_{\ell,k}\stackrel{\text{def}}{=}Y_{\ell,k}-X_{\ell,k}
 \end{equation}
 for each $\ell\in\mathcal{L}$ and each $k\in\{1, 2, \ldots, n\}$, we obtain
\begin{align}
&\Pr_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}}\left\{\log\bigg(\frac{\prod_{k=1}^n q_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{Y}_k|\boldsymbol{X}_{k})}{r_{\boldsymbol{Y}^n}(\boldsymbol{Y}^n)}\bigg) \ge \log \xi_n \right\}\notag\\*
& \le \Pr_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{*}}\left\{n\mathrm{C}(\mathbf{P}^*)+ \sum_{k=1}^n\sum_{\ell=1}^L\frac{-\big(\frac{P_\ell}{N_\ell} \big) Z_{\ell,k}^2 + 2X_{\ell,k}Z_{\ell,k}+X_{\ell,k}^2}{2(P_\ell+N_\ell)} \ge \log \xi_n - \log 2\right\} \notag\\*
&\qquad + \sum_{\mathbf{s}\in \mathcal{S}^{(n)}\setminus\Pi^{(n)}}\Pr_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{(\mathbf{s})}}\left\{n\mathrm{C}(\mathbf{s})+\sum_{k=1}^n\sum_{\ell=1}^L\frac{-\big(\frac{s_\ell}{N_\ell}\big) Z_{\ell,k}^2 + 2X_{\ell,k}Z_{\ell,k}+X_{\ell,k}^2}{2(s_\ell+N_\ell)} \ge \log \xi_n - \log\big(2|\mathcal{S}^{(n)}|\big)\right\} \label{eqnBHT5th1Chain}
\end{align}
where $\mathrm{C}(\cdot)$ is as defined in~\eqref{defCPparallelNFB}.
In order to simplify the RHS of~\eqref{eqnBHT5th1Chain}, we define $\xi_n>0$ such that
\begin{align}
\log \xi_n \stackrel{\text{def}}{=} n\mathrm{C}(\mathbf{P}^*) + \sqrt{n}\left(\sqrt{\mathrm{V}(\mathbf{P}^*)}\, \Phi^{-1}(\varepsilon+\tau)\right) + \log\big(2|\mathcal{S}^{(n)}|\big). \label{defxin}
\end{align}
In addition, for each $\mathbf{d}\in \mathbb{R}_+^L$, let
\begin{align}
U_k^{(\mathbf{d})} \stackrel{\text{def}}{=}  \sum_{\ell=1}^L\frac{-\big(\frac{d_\ell}{N_\ell}\big) Z_{\ell,k}^2 + 2X_{\ell,k}Z_{\ell,k}+d_\ell}{2(d_\ell+N_\ell)} \label{defUkd}
\end{align}
for each $k\in\{1, 2, \ldots, n\}$.
By using~\eqref{eqnBHT5th1Chain}, \eqref{defxin} and \eqref{defUkd} together with the facts by~\eqref{modifiedCodeProperty} that % and~\eqref{pEllValue*}
\begin{align}
\Pr_{ p_{\boldsymbol{X}^n,\boldsymbol{Y}^n}^*}\left\{\left\{\phi(\boldsymbol{X}^n)\in\Gamma^{(L^2\gamma)}(\mathbf{P}^*)\right\}\cap\bigg\{\sum\limits_{\ell=1}^{L}\sum\limits_{k=1}^{n}X_{\ell, k}^2  = nP\bigg\}\right\}=1 \label{defDistP*}
\end{align}
and
\begin{align}
\Pr_{ p_{\boldsymbol{X}^n,\boldsymbol{Y}^n}^{(\mathbf{s})}}\left\{\phi(\boldsymbol{X}^n)\in\Gamma^{(0)}(\mathbf{s})\right\}=1 \label{defDistPs}
\end{align}
for each $\mathbf{s}\in \mathcal{S}^{(n)}$,
we can express \eqref{eqnBHT5th1Chain} as
\begin{align}
&\Pr_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}}\left\{\log\bigg(\frac{\prod_{k=1}^n q_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{Y}_k|\boldsymbol{X}_{k})}{r_{\boldsymbol{Y}^n}(\boldsymbol{Y}^n)}\bigg) \ge \log \xi_n \right\}\notag\\*
&\quad\le \Pr_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{*}}\left\{\frac{1}{\sqrt{n \mathrm{V}(\mathbf{P}^*)}}\sum_{k=1}^n U_k^{(\mathbf{P}^*)}  \ge \Phi^{-1}(\varepsilon+\tau)\right\} \notag\\
&\qquad + \sum_{\mathbf{s}\in \mathcal{S}^{(n)}\setminus\Pi^{(n)}}\Pr_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{(\mathbf{s})}}\left\{\frac{1}{\sqrt{n}}\sum_{k=1}^n U_k^{(\mathbf{s})}\ge \sqrt{n}\big(\mathrm{C}(\mathbf{P}^*)-\mathrm{C}(\mathbf{s})\big)+ \sqrt{\mathrm{V}(\mathbf{P}^*)}\, \Phi^{-1}(\varepsilon+\tau)\right\}. \label{eqnBHT5th2Chain}
\end{align}
\subsection{Applying Curtiss' Theorem When $\phi(\boldsymbol{X}^n)$ is Close to $\mathbf{P}^*$} \label{stepBInProof}
In order to simplify the first term in~\eqref{eqnBHT5th2Chain}, we
define
%\begin{align}
%V_k^{(\mathbf{d})} \stackrel{\text{def}}{=}  \sum_{\ell=1}^L\frac{-\big(\frac{d_\ell}{N_\ell}\big) Z_{\ell,k}^2 + 2\sqrt{d_\ell}Z_{\ell,k}+d_\ell}{2(d_\ell+N_\ell)} \label{defVkd}
%\end{align}
%for all $\mathbf{d}\in \mathbb{R}_+^L$
\begin{align}
V_k^{(\mathbf{P}^*)} \stackrel{\text{def}}{=}  \sum_{\ell=1}^L\frac{-\big(\frac{P_\ell}{N_\ell}\big) Z_{\ell,k}^2 + 2\sqrt{P_\ell}Z_{\ell,k}+P_\ell}{2(P_\ell+N_\ell)} \label{defVkd}
\end{align}
for each $k\in\{1, 2, \ldots, n\}$
and want to show that
\begin{align}
\lim_{n\rightarrow \infty}\E_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{*}}\left[e^{\frac{t}{\sqrt{n }}\sum\limits_{k=1}^n U_k^{(\mathbf{P}^*)}}\right] = \lim_{n\rightarrow \infty}\E_{p_{\boldsymbol{Z}^n}^{*}}\left[e^{\frac{t}{\sqrt{n }}\sum\limits_{k=1}^n V_k^{(\mathbf{P}^*)}}\right] \label{levyThmEq1}
\end{align}
for all $t\in \mathbb{R}$ where
\begin{align}
p_{\boldsymbol{Z}^n}^{*}(\mathrm{z}^n) \triangleq \prod_{k=1}^n \mathcal{N} (z_{\ell,k}; 0, N_\ell).
\end{align}
 To this end, recall the following statements due to the channel law: %that for each $n\in\{L+2, L+3, \ldots\}$
 \begin{enumerate}
 \item[(i)] $Z_{\ell,k}\sim\mathcal{N}(z_{\ell,k};0, N_\ell)$ for all $\ell\in \mathcal{L}$ and all $k\in\{1, 2, \ldots, n\}$;
 \item[(ii)] $\{Z_{\ell,k}|\ell\in\mathcal{L}, k\in\{1, 2, \ldots, n\}\}$ are independent;
 \item[(iii)] $\boldsymbol{Z}_k$ and $(\boldsymbol{X}^{k},\boldsymbol{Y}^{k-1},\boldsymbol{Z}^{k-1})$ are independent for all $k\in\{1, 2, \ldots, n\}$.
 \end{enumerate}
For any $t\in\mathbb{R}$ and any~$n\in\{L+2, L+3, \ldots\}$ such that $n\ge t^2$, since
\begin{align}
P_\ell - L^2\gamma-\frac{1}{n}\sum\limits_{k=1}^n X_{\ell,k}^2 \le 0 \le P_\ell + L^2\gamma-\frac{1}{n}\sum\limits_{k=1}^n X_{\ell,k}^2
\end{align}
by~\eqref{defDistP*} and $P_\ell+N_\ell+\frac{tP_\ell}{\sqrt{n}} >0$
for all $\ell\in\mathcal{L}$, we have
 \begin{align}
& \E_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{*}}\Biggl[e^{\frac{t}{\sqrt{n}}\sum\limits_{k=1}^n U_k^{(\mathbf{P}^*)}}\cdot e^{t^2\sum\limits_{\ell=1}^L\frac{N_\ell \left(P_\ell - L^2\gamma-\frac{1}{n}\sum\limits_{k=1}^n X_{\ell,k}^2\right) }{2(P_\ell+N_\ell)\left(P_\ell+N_\ell+\frac{tP_\ell}{\sqrt{n }}\right)}}\:\Biggr] \notag\\*
 &\le \E_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{*}}\left[e^{\frac{t}{\sqrt{n }}\sum\limits_{k=1}^n U_k^{(\mathbf{P}^*)}}\right] \\*
  &\le \E_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{*}}\Biggl[e^{\frac{t}{\sqrt{n}}\sum\limits_{k=1}^n U_k^{(\mathbf{P}^*)}}\cdot e^{t^2\sum\limits_{\ell=1}^L\frac{N_\ell \left(P_\ell + L^2\gamma-\frac{1}{n}\sum\limits_{k=1}^n X_{\ell,k}^2\right) }{2(P_\ell+N_\ell)\left(P_\ell+N_\ell+\frac{tP_\ell}{\sqrt{n }}\right)}}\:\Biggr]. \label{levyThmEq1*}
 \end{align}
In order to simplify the above chain of inequalities, we need the following lemma, whose proof is deferred to Appendix~\ref{appendixC} because it involves straightforward calculations based on~\eqref{defUkd}, \eqref{defVkd} and the channel law.
 \smallskip
 \begin{Lemma} \label{lemma1InProof}
 For any $\lambda\in\mathbb{R}$, we have
 \begin{align}
 \E_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{*}}\Biggl[e^{\lambda\sum\limits_{k=1}^n U_k^{(\mathbf{P}^*)}}\cdot e^{\lambda^2\sum\limits_{\ell=1}^L\frac{N_\ell \left(nP_\ell  -\sum\limits_{k=1}^n X_{\ell,k}^2\right) }{2(P_\ell+N_\ell)\left((1+\lambda)P_\ell+N_\ell\right)}}\:\Biggr] = \E_{p_{\boldsymbol{Z}^n}^{*}}\left[e^{\lambda\sum\limits_{k=1}^n V_k^{(\mathbf{P}^*)}}\right]. \label{appendixCeq1}
 \end{align}
 \end{Lemma}

Lemma~\ref{lemma1InProof}, which forms the crux of the proof of Theorem \ref{thmMainResult}, is important because it
 establishes the equivalence in distribution between the sum $\sum_{k=1}^n U_k^{(\mathbf{P}^*)}$, which contains {\em dependent} random variables, and the sum $\sum_{k=1}^n V_k^{(\mathbf{P}^*)}$, which contains {\em independent} random variables. The former is intractable to analyze while the latter can be analyzed in a straightforward manner by invoking the central limit theorem.
 %
% establishes the equivalence between the moment generating functions of the dependent sum  and independent sum .


Using Lemma~\ref{lemma1InProof}, we can simplify~\eqref{levyThmEq1*} through the identification $\lambda\equiv \frac{t}{\sqrt{n}}$ and obtain
\begin{align}
&\E_{p_{\boldsymbol{Z}^n}^{*}}\left[e^{\frac{t}{\sqrt{n }}\sum\limits_{k=1}^n V_k^{(\mathbf{P}^*)}}\right] \cdot e^{-t^2 L^2\gamma \sum\limits_{\ell=1}^L\frac{N_\ell }{2(P_\ell+N_\ell)\left(P_\ell+N_\ell+\frac{tP_\ell}{\sqrt{n }}\right)}}  \notag\\*
&\le \E_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{*}}\left[e^{\frac{t}{\sqrt{n }}\sum\limits_{k=1}^n U_k^{(\mathbf{P}^*)}}\right]
\\*
&\le \E_{p_{\boldsymbol{Z}^n}^{*}}\left[e^{\frac{t}{\sqrt{n }}\sum\limits_{k=1}^n V_k^{(\mathbf{P}^*)}}\right] \cdot e^{t^2 L^2\gamma \sum\limits_{\ell=1}^L\frac{  N_\ell }{2(P_\ell+N_\ell)\left(P_\ell+N_\ell+\frac{tP_\ell}{\sqrt{n }}\right)}}.
\label{levyThmEq2}
\end{align}
%For the sake of completeness, the derivation of~\eqref{levyThmEq2} can be found in Appendix~\ref{appendixC}.
Combining \eqref{levyThmEq2} and~\eqref{defgammaInProof}, we conclude that \eqref{levyThmEq1} holds for each $t\in\mathbb{R}$. Since the moment generating functions of $\frac{1}{\sqrt{n }}\sum_{k=1}^n V_k^{(\mathbf{P}^*)}$ and $\frac{1}{\sqrt{n }}\sum_{k=1}^n U_k^{(\mathbf{P}^*)}$ converge to the same function, it follows from Curtiss' theorem~\cite[Th.~3]{curtiss} (as stated in Theorem~\ref{thmCurtiss}) that
\begin{align}
\lim_{n\rightarrow \infty} \Pr_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{*}}\left\{\frac{1}{\sqrt{n \mathrm{V}(\mathbf{P}^*)}}\sum_{k=1}^n U_k^{(\mathbf{P}^*)}  \ge \Phi^{-1}(\varepsilon+\tau)\right\} = \lim_{n\rightarrow \infty} \Pr_{p_{\boldsymbol{Z}^n}^{*}}\left\{\frac{1}{\sqrt{n \mathrm{V}(\mathbf{P}^*)}}\sum_{k=1}^n V_k^{(\mathbf{P}^*)}  \ge \Phi^{-1}(\varepsilon+\tau)\right\}. \label{levyThmEq3}
\end{align}
Recognizing that $\big\{V_k^{(\mathbf{P}^*)} \big\}_{k=1}^\infty$ are independent zero-mean Gaussian random variables with variance $\mathrm{V}(\mathbf{P}^*)$ by the definition of $V_k^{(\mathbf{P}^*)} $ in~\eqref{defVkd} and the definition of $\mathrm{V}(\mathbf{P}^*)$ in~\eqref{defVP}, we apply the central limit theorem~\cite{feller} and obtain
\begin{align}
\lim_{n\rightarrow \infty} \Pr_{p_{\boldsymbol{Z}^n}^{*}}\left\{\frac{1}{\sqrt{n \mathrm{V}(\mathbf{P}^*)}}\sum_{k=1}^n V_k^{(\mathbf{P}^*)} \le \Phi^{-1}(\varepsilon+\tau)\right\} = \varepsilon+\tau,
\end{align}
which together with~\eqref{levyThmEq3} implies that
\begin{align}
\lim_{n\rightarrow \infty} \Pr_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{*}}\left\{\frac{1}{\sqrt{n \mathrm{V}(\mathbf{P}^*)}}\sum_{k=1}^n U_k^{(\mathbf{P}^*)}  \ge \Phi^{-1}(\varepsilon+\tau)\right\} = 1-\varepsilon - \tau. \label{levyThmEq4}
\end{align}
\subsection{Applying Large Deviation Bounds When $\phi(\boldsymbol{X}^n)$ is Far from $\mathbf{P}^*$} \label{stepCInProof}
In order to bound the second term in~\eqref{eqnBHT5th2Chain}, we consider a fixed~$n\in\{L+2, L+3, \ldots\}$ and want to show that there exists some $\kappa>0$ such that
\begin{align}
\mathrm{C}(\mathbf{P}^*)-\mathrm{C}(\mathbf{s}) \ge \kappa\|\mathbf{P}^*-\mathbf{s}\|_{2}^2 \label{eqnBHT6thChain}
\end{align}
for all $\mathbf{s}\in \mathcal{S}^{(n)}$. To this end, we first define the Lagrangian function $f: \mathbb{R}^L \rightarrow \mathbb{R}$ as
\begin{align}
f(\mathbf{d}) \stackrel{\text{def}}{=}   \mathrm{C}(\mathbf{d}) - \frac{1}{2\Lambda}\left(\sum_{\ell=1}^L d_\ell - P\right) + \sum_{\ell=1}^L \mu_\ell d_\ell \label{defFunctionF}
\end{align}
where $\Lambda\ge 0$ is the unique number that satisfies~\eqref{sumPell=P} and~\eqref{pEllValue} and $\mu_\ell\ge 0$ is defined for each $\ell\in\mathcal{L}$ as
\begin{align}
\mu_\ell \stackrel{\text{def}}{=}
\begin{cases}
0 & \text{if $P_\ell>0$,}\\
\frac{1}{2\Lambda}-\frac{1}{2N_\ell} & \text{if $P_\ell=0$.}
\end{cases} \label{defMu}
\end{align}
 Define $N_\text{max} \stackrel{\text{def}}{=} \max\limits_{\ell\in\mathcal{L}}N_\ell$. Then for all $\mathbf{s}\in \mathcal{S}^{(n)}$, we use Taylor's theorem to obtain
\begin{align}
 f(\mathbf{s}) = f(\mathbf{P}^*) + (\mathbf{s}-\mathbf{P}^*)^t\triangledown f(\mathbf{P}^*) +  \frac{1}{2}(\mathbf{s}-\mathbf{P}^*)^t\triangledown^2 f(\bar{\mathbf{s}}) (\mathbf{s}-\mathbf{P}^*) \label{eqnTaylor}
\end{align}
for some $\bar{\mathbf{s}}$ that lies on the line that connects $\mathbf{s}$ and $\mathbf{P}^*$, where $\triangledown f(\mathbf{P}^*)$ denotes the gradient which satisfies
\begin{align}
\triangledown f(\mathbf{P}^*) = 0 \label{eqnTaylor1st}
\end{align}
and $\triangledown^2 f(\bar{\mathbf{s}})$ denotes the Hessian matrix that satisfies
\begin{align}
(\mathbf{s}-\mathbf{P}^*)^t\triangledown^2 f(\bar{\mathbf{s}}) (\mathbf{s}-\mathbf{P}^*) \le -\frac{\|\mathbf{s}-\mathbf{P}^*\|_2^2}{2 (N_{\max} +P)^2 }. \label{eqnTaylor2nd}
\end{align}
For the sake of completeness, the derivations of~\eqref{eqnTaylor1st} and~\eqref{eqnTaylor2nd} are contained in Appendix~\ref{appendixB}.
Combining~\eqref{eqnTaylor}, \eqref{eqnTaylor1st} and~\eqref{eqnTaylor2nd}, we have for all $\mathbf{s}\in \mathcal{S}^{(n)}$
\begin{align}
f(\mathbf{P}^*)-f(\mathbf{s}) \ge \frac{\|\mathbf{s}-\mathbf{P}^*\|_2^2}{4 (N_{\max} +P)^2 },
\end{align}
which together with the definitions of~$f$ and~$\mu_\ell$ in~\eqref{defFunctionF} and~\eqref{defMu} respectively implies that
\begin{align}
\mathrm{C}(\mathbf{P}^*)-\mathrm{C}(\mathbf{s}) &\ge \frac{\|\mathbf{s}-\mathbf{P}^*\|_2^2}{4 (N_{\max} +P)^2 } + \sum_{\ell=1}^L\mu_\ell(s_\ell-P_\ell) \\*
& \ge  \frac{\|\mathbf{s}-\mathbf{P}^*\|_2^2}{4 (N_{\max} +P)^2 }.
\end{align}
Consequently, \eqref{eqnBHT6thChain} holds by setting
\begin{equation}
\kappa\stackrel{\text{def}}{=} \frac{1}{4 (N_{\max} +P)^2}. \label{defKappa}
\end{equation}
Following~\eqref{eqnBHT5th2Chain}, we consider for each $\mathbf{s}\in \mathcal{S}^{(n)}\setminus\Pi^{(n)}$
\begin{align}
&\Pr_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{(\mathbf{s})}}\left\{\frac{1}{\sqrt{n}}\sum_{k=1}^n U_k^{(\mathbf{s})}\ge \sqrt{n}\big(\mathrm{C}(\mathbf{P}^*)-\mathrm{C}(\mathbf{s})\big)+ \sqrt{\mathrm{V}(\mathbf{P}^*)}\, \Phi^{-1}(\varepsilon+\tau)\right\}\notag\\
&\le \Pr_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{(\mathbf{s})}}\left\{\frac{1}{\sqrt{n}}\sum_{k=1}^n U_k^{(\mathbf{s})}\ge \kappa\sqrt{n}\|\mathbf{P}^*-\mathbf{s}\|_{2}^2+ \sqrt{\mathrm{V}(\mathbf{P}^*)}\, \Phi^{-1}(\varepsilon+\tau)\right\} \label{eqnBHT7thChaina}\\
&\le \Pr_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{(\mathbf{s})}}\left\{\frac{1}{\sqrt{n}}\sum_{k=1}^n U_k^{(\mathbf{s})}\ge \kappa n^{1/6}+ \sqrt{\mathrm{V}(\mathbf{P}^*)}\, \Phi^{-1}(\varepsilon+\tau)\right\} \label{eqnBHT7thChainb}
\end{align}
where
\begin{itemize}
\item \eqref{eqnBHT7thChaina} is due to~\eqref{eqnBHT6thChain}.
\item \eqref{eqnBHT7thChainb} follows from the definition of~$\Pi^{(n)}$ in~\eqref{defSetPiN}.
\end{itemize}
Following the standard approach for obtaining large deviation bounds, we apply Markov's inequality on the RHS of~\eqref{eqnBHT7thChainb} and obtain for each $\mathbf{s}\in \mathcal{S}^{(n)}\setminus\Pi^{(n)}$
\begin{align}
&\Pr_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{(\mathbf{s})}}\left\{\frac{1}{\sqrt{n}}\sum_{k=1}^n U_k^{(\mathbf{s})}\ge \sqrt{n}\big(\mathrm{C}(\mathbf{P}^*)-\mathrm{C}(\mathbf{s})\big)+ \sqrt{\mathrm{V}(\mathbf{P}^*)}\, \Phi^{-1}(\varepsilon+\tau)\right\}\notag\\
&\le \frac{\E_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{(\mathbf{s})}}\left[e^{\frac{1}{\sqrt{n}}\sum_{k=1}^n U_k^{(\mathbf{s})}}\right]}{e^{\kappa n^{1/6}+ \sqrt{\mathrm{V}(\mathbf{P}^*)}\, \Phi^{-1}(\varepsilon+\tau)}}. \label{eqnBHT8thChain}
\end{align}
In order to bound the RHS of~\eqref{eqnBHT8thChain}, consider the following chain of inequalities for each $\mathbf{s}\in \mathcal{S}^{(n)}\setminus\Pi^{(n)}$:
\begin{align}
\E_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{(\mathbf{s})}}\left[e^{\frac{1}{\sqrt{n}}\sum_{k=1}^n U_k^{(\mathbf{s})}}\right]& = \left(\prod_{\ell=1}^L\frac{s_\ell+N_\ell}{(1+n^{-1/2})s_\ell+N_\ell}\right)^{n/2}e^{\sum\limits_{\ell=1}^L\left(\frac{\sqrt{n}s_\ell}{2(s_\ell+N_\ell)}+\frac{N_\ell s_\ell}{2(s_\ell+N_\ell)\left(\left(1+n^{-1/2}\right)s_\ell+N_\ell\right)}\right)}\label{eqnBHT9thChaina}\\
  &\le \left(\prod_{\ell=1}^L\left(1-\frac{n^{-1/2}s_\ell}{(1+n^{-1/2})s_\ell+N_\ell}\right)^{n/2}e^{\frac{\sqrt{n}s_\ell}{2(s_\ell+N_\ell)}}\right)e^{\sum\limits_{\ell=1}^L\frac{N_\ell s_\ell}{2(s_\ell+N_\ell)^2}} \\
& \le e^{\sum\limits_{\ell=1}^L\left( \frac{s_\ell^2}{2((1+n^{-1/2})s_\ell+N_\ell)(s_\ell+N_\ell)} +\frac{N_\ell s_\ell}{2(s_\ell+N_\ell)^2}\right)} \label{eqnBHT9thChainb}\\
&\le e^{\sum\limits_{\ell=1}^L\frac{ s_\ell}{2(s_\ell+N_\ell)}} \\*
&\le e^{L/2},\label{eqnBHT9thChain}
\end{align}
where
\begin{itemize}
\item \eqref{eqnBHT9thChaina} follows from straightforward calculations based on the definition of $U_k^{(\mathbf{s})}$ in~\eqref{defUkd}, the property of~$p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{(\mathbf{s})}$ in~\eqref{defDistPs} and the channel law, which are elaborated in Appendix~\ref{appendixD} for the sake of completeness.
\item \eqref{eqnBHT9thChainb} is due to the fact that $(1-\frac{1}{x})^x \le e^{-1}$ for all $x>1$.
\end{itemize}
Combining~\eqref{eqnBHT8thChain} and~\eqref{eqnBHT9thChain}, we have the following large deviation bound for each $\mathbf{s}\in \mathcal{S}^{(n)}\setminus\Pi^{(n)}$:
\begin{align}
\Pr_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{(\mathbf{s})}}\left\{\frac{1}{\sqrt{n}}\sum_{k=1}^n U_k^{(\mathbf{s})}\ge \sqrt{n}\big(\mathrm{C}(\mathbf{P}^*)-\mathrm{C}(\mathbf{s})\big)+ \sqrt{\mathrm{V}(\mathbf{P}^*)}\, \Phi^{-1}(\varepsilon+\tau)\right\}\le\frac{e^{L/2}}{e^{\kappa n^{1/6}+ \sqrt{\mathrm{V}(\mathbf{P}^*)}\, \Phi^{-1}(\varepsilon+\tau)}}. \label{eqnBHT9*thChain}
\end{align}
Following~\eqref{eqnBHT5th2Chain}, we use \eqref{eqnBHT9*thChain} and~\eqref{cardinalityBound} to obtain
\begin{align}
\lim_{n\rightarrow\infty}\sum_{\mathbf{s}\in \mathcal{S}^{(n)}\setminus\Pi^{(n)}}\Pr_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{(\mathbf{s})}}\left\{\frac{1}{\sqrt{n}}\sum_{k=1}^n U_k^{(\mathbf{s})}\ge \sqrt{n}\big(\mathrm{C}(\mathbf{P}^*)-\mathrm{C}(\mathbf{s})\big)+ \sqrt{\mathrm{V}(\mathbf{P}^*)}\, \Phi^{-1}(\varepsilon+\tau)\right\}=0. \label{eqnBHT10thChain}
\end{align}
\subsection{Combining Earlier Bounds to Obtain an Upper Bound on~$\mathrm{L}_\varepsilon^{\text{fb}}$}
Combining~\eqref{eqnBHTSecondChain}, \eqref{defxin}, \eqref{eqnBHT5th2Chain}, \eqref{levyThmEq4}, \eqref{eqnBHT10thChain} and~\eqref{cardinalityBound}, we have
\begin{align}
\log M_{\mathrm{fb}}^*(\bar n, \varepsilon, P)
 \le n\mathrm{C}(\mathbf{P}^*) + \sqrt{n}\left(\sqrt{\mathrm{V}(\mathbf{P}^*)}\, \Phi^{-1}(\varepsilon+\tau)\right) + \log\big(2n^L\big) - \log\left(\tau/2\right)  \label{eqnBHT11thChain}
\end{align}
for all sufficiently large~$n$, which together with~\eqref{defBarN} implies that
\begin{equation}
\liminf\limits_{\bar n\rightarrow \infty}\frac{1}{\sqrt{\bar n}}\big(\log M_{\mathrm{fb}}^*(\bar n, \varepsilon, P)-\bar n\mathrm{C}(\mathbf{P}^*)\big) \le \sqrt{\mathrm{V}(\mathbf{P}^*)}\, \Phi^{-1}(\varepsilon+ \tau). \label{eqnBHT12thChain}
\end{equation}
Since $\tau>0$ is arbitrary, it follows from~\eqref{eqnBHT12thChain} and Definition~\ref{defDispersion} that
\begin{equation}
\mathrm{L}_\varepsilon^{\text{fb}}\le \sqrt{\mathrm{V}(\mathbf{P}^*)}\, \Phi^{-1}(\varepsilon).
\end{equation}
\section{Concluding Remarks} \label{sectionConclusion}
\subsection{Novel Ingredients in Proof of Theorem~\ref{thmMainResult}} \label{sectionNovelIngredients}
As mentioned in Section~\ref{subsecRelatedWork}, the proof of~\cite[Th.~ 2]{AW14} which obtains upper bounds on the second-order asymptotics of DMCs with feedback cannot be generalized to the parallel Gaussian channel with feedback. Indeed, the proof of Theorem~\ref{thmMainResult} follows the standard procedures for obtaining the second-order asymptotics of DMCs without feedback (see, e.g., \cite[proof of Th.~50]{Pol10} and~\cite[Sec.~III]{TomTan12}) except the following three novel ingredients:
\begin{enumerate}
\item Instead of classifying transmitted codewords into polynomially many type classes based on their empirical distributions which is generally not possible for channels with continuous input alphabet, we discretize the transmitted power and classify the codewords into polynomially many type classes based on their discretized power types. In particular, the collection of \emph{power type classes} $\mathcal{S}^{(n)}$ in~\eqref{defSetS} plays a key role in our analysis, and there are polynomially many power type classes by~\eqref{cardinalityBound}.
    %The first step of the proof is to quantize the power type of the transmitted codeword up to a precision level of $P/n$ so that
     %quantize their power types up to a precision level of $P/n$.
    The details can be found in Section~\ref{stepAinProof} in the proof.
\item Curtiss' theorem rather than Berry-Ess\'een theorem is invoked to bound the information spectrum term (the first term in~\eqref{eqnBHT5th2Chain}) related to transmitted codewords whose types are close to the optimal power allocation. In particular, Berry-Ess\'een theorem for bounded martingale difference sequences cannot be used to bound the information spectrum term in the presence of feedback because each input symbol~$X_{\ell,k}$ belongs to an interval $[-\sqrt{nP}, \sqrt{nP}]$ that grows unbounded as~$n$ increases. Instead, we apply Curtiss' theorem to show that the distribution of the sum of random variables in the information spectrum term converges to some distribution generated from a sum of i.i.d.\ random variables (i.e., \eqref{levyThmEq1}), thus facilitating the use of the usual central limit theorem~\cite{feller}. The details can be found in Section~\ref{stepBInProof}.
    \item In order to bound the information spectrum term related to transmitted codewords whose types are far from the optimal power allocation (the second term in~\eqref{eqnBHT5th2Chain}), the usual approach is to bound the information spectrum term by an \emph{average } of exponentially many upper bounds where each upper bound is then further simplified by invoking Chebyshev's inequality~\cite[Sec.~X-A]{Hayashi09}. However, due to the presence of feedback, the information spectrum term can be expressed as only a sum (instead of average) of polynomially many upper bounds as shown in the second term in~\eqref{eqnBHT5th2Chain}. In order to control the \emph{sum} of polynomially many upper bounds, we have to resort to large deviation bounds as shown in~\eqref{eqnBHT9*thChain} rather than the weaker Chebyshev's inequality. The details can be found in Section~\ref{stepCInProof}.
\end{enumerate}

%A more challenging extension is to characterize the second-order asymptotics of any DMC with feedback whose corresponding capacity-achieving input distributions are not unique, where some partial results were reported in~\cite{AW14}.}
\subsection{Major Difficulties in Tightening the Third-Order Term}
If the feedback link is absent, the third-order term of the optimal finite blocklenth rate $\frac{1}{n}\log M_{\text{fb}}^*(n,\varepsilon,P)$ is $\Theta\Big(\frac{\log n}{n}\Big)$ as shown in~\eqref{eqn:asymp_expans} in Section~\ref{Introduction}.
% the optimal rate  satisfies
%\begin{equation}
%\frac{1}{n}\log M_{\text{fb}}^*(n,\varepsilon,P)\ge \mathrm{C}(\mathbf{P}^*) + \sqrt{\frac{\mathrm{V}(\mathbf{P}^*)}{n}}\, \Phi^{-1} (\varepsilon) + .
%\end{equation}
The third-order term can be obtained by applying Berry-Ess\'een theorem to bound an information spectrum term (analogous to the first term in~\eqref{eqnBHT5th2Chain}).

In the presence of feedback, Theorem~\ref{thmMainResult} shows that the third-order term is $o\Big(\frac{1}{\sqrt{n}}\Big)$. If we want to compute an explicit upper bound on the third-order term using the current proof technique, an intuitive way is to invoke a non-asymptotic version of Curtiss' theorem that can measure the proximity between two distributions based on the proximity between their moment generating functions. However, such a non-asymptotic version of Curtiss' theorem does not exist to the best of our knowledge, which prohibits us from strengthening the current $o\Big(\frac{1}{\sqrt{n}}\Big)$ bound on the third-order term.
It is worth noting that~\eqref{levyThmEq1*} and~\eqref{levyThmEq2} in our proof break down if the moment generating functions are replaced with characteristic functions. If one can find a way to make characteristic functions amenable to our proof approach, then
a non-asymptotic version of L\'evy's continuity theorem known as \emph{Ess\'een's smoothing lemma} (see, e.g., \cite[Th.~1.5.2]{IbragimovLinnik1971}) may be invoked to tighten the third-order term herein.
\subsection{Future Work} \label{sectionFutureWork}
%An extension of this work may use the aforementioned ingredients to derive the second-order asymptotics of any DMC with feedback under a cost constraint whose corresponding capacity-achieving input distribution is unique.
The techniques presented herein may be used to analyze the fixed-error asymptotics of fixed-length codes over parallel DMCs with cost constraint and with feedback. We envision that there will be an added layer of complexity as the {\em method of types}~\cite{Csi97} is typically used to analyze the fixed-error asymptotics of DMCs with and without cost constraint~\cite{Strassen}. Hence, we anticipate that {\em two} applications of the method of types need to be applied --- one for handling power types that specify the power allocation among the parallel channels (as was done in Section~\ref{stepAinProof}) and another for handling codewords of the same power type but of different empirical distributions (the usual types).
 In the present work, the latter situation is ameliorated by the fact that the maximum rate achievable by a coding scheme over an AWGN channel is completely determined by the power allocated for the coding scheme.
%\subsection{Simplifying the Non-Asymptotic Bound}
%The channel law is
%\begin{equation}
%q_{\boldsymbol{Y}|\boldsymbol{X}}(y_{k}|x_{k}) = \phi_{0,1}(y_k-x_k) \label{memorylessStatement++}
%\end{equation}
%for each $k\in\{1, 2, \ldots, n\}$.
%Combining \eqref{memorylessStatement++} and Property~(iii) in Lemma~\ref{lemmaSimulatingDistribution}, we have
%\begin{equation}
%\log\frac{q_{\boldsymbol{Y}|\boldsymbol{X}}(Y_{k}|X_{k})}{s_{Y_{k}}(Y_{k})} = \frac{1}{2}\log(1+P)+ \frac{}{2(1+P)}\left(-P(\boldsymbol{Y}_k-\boldsymbol{X}_k)^2 + \boldsymbol{X}_k^2 + 2\boldsymbol{X}_k(\boldsymbol{Y}_k-\boldsymbol{X}_k)\right) \label{defLogLikelihood}
%\end{equation}
%for each $k\in\{1, 2, \ldots, n\}$. Due to the power equality constraint imposed on the codewords, we have
%\begin{equation}
%\Pr_{p_{W, X^n, Y^n}}\left\{ \sum_{k=1}^n  \boldsymbol{X}_k^2 = nP \right\} \stackrel{\eqref{powerConstraintInProofnBar}}{=}1. \label{powerConstraintInProofn}
%\end{equation}
% Letting
% \begin{equation}
%U_k \stackrel{\text{def}}{=} \frac{}{2(1+P)}(-P(\boldsymbol{Y}_k-\boldsymbol{X}_k)^2 + 2\boldsymbol{X}_k(\boldsymbol{Y}_k-\boldsymbol{X}_k)+P) \label{defUk}
% \end{equation}
% for each $k\in\{1, 2, \ldots, n\}$, it follows from \eqref{defLogLikelihood} and \eqref{powerConstraintInProofn} that
%\begin{equation}
%\Pr_{p_{W, X^n, Y^n}}\left\{\sum_{k=1}^n \log\frac{q_{\boldsymbol{Y}|\boldsymbol{X}}(Y_{k}|X_{k})}{s_{Y_{k}}(Y_{k})} =\frac{n}{2}\log(1+P) + \sum_{k=1}^n U_k\right\}=1.\label{expectationSumUk}
%\end{equation}
%Combining \eqref{eqnBHTSecondChain} and \eqref{expectationSumUk}, we have
%\begin{align}
%\log M_{\mathrm{fb}}^*(n-1, \varepsilon, P) \le \log\xi_n - \log\left(\Pr_{p_{W, X^n, Y^n}} \left\{\sum_{k=1}^n U_k < \log\xi_n - \frac{n}{2}\log(1+P) \right\}-\varepsilon\right). %\label{eqnBHT6thChain}
%\end{align}
%%It can be shown from \eqref{memorylessStatement*} and \eqref{defUk} that
%%\[
%%\E_{p_{W,X^n, Y^n}}\left[ U_k |X^{k-1}, \boldsymbol{Y}^{k-1}\right] =0
%%\]
%%for each $k\in\{1, 2, \ldots, n\}$, which implies that $\sum_{k=1}^n U_k$ is a martingale.
%\subsection{Evaluating the Distribution of the Sum of Random Variables $\sum_{k=1}^n U_k$}
%In order to simplify \eqref{eqnBHT6thChain}, we now investigate the distribution of the sum of random variables $\sum_{k=1}^n U_k$. Note that if the AWGN channel has no feedback, it follows from spherical symmetry \cite[Section 4.2.2]{Pol10} of the AWGN channel   that the evaluation of $\eqref{eqnBHT6thChain}$ can be simplified by assuming without loss of generality that
%\begin{equation}
%(X_1, X_2, \ldots, X_n) = (\sqrt{P}, \sqrt{P}, \ldots, \sqrt{P}). \label{powerAssumption}
% \end{equation}
%Surprisingly in the feedback case, we will show in the following that the distribution of $\sum_{k=1}^n U_k$ can be evaluated in closed form.  We need not  appeal to any sophisticated Berry-Ess\'een-type results for bounded martingale difference sequences~\cite{Machkouri} as was done by Altu\u{g} and Wagner for discrete memoryless channels in \cite{AW14}.
%% not affected by imposing the extra assumption \eqref{powerAssumption} and
%The evaluation of $\eqref{eqnBHT6thChain}$ is as simple as the no-feedback case. Define the function
%$
%\lambda: \mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}
%$
%\begin{equation}
%\lambda(x, y)= -P(y-x)^2 + 2 x(y-x). \label{defLambda}
%\end{equation}
%We begin evaluating the distribution of $\sum_{k=1}^n U_k$ by examining the distribution of
%$\sum_{k=1}^n \lambda(\boldsymbol{X}_k, \boldsymbol{Y}_k)$ (cf.\ \eqref{defUk}) as follows.
%Let
%\begin{equation}
%\E_{p_{W, X^n, Y^n}}\left[e^{t\sum_{k=1}^n \lambda(\boldsymbol{X}_k, \boldsymbol{Y}_k)}\right] \label{momentGenPartialUk}
%\end{equation}
% be the moment generating function of $\sum_{k=1}^n \lambda(\boldsymbol{X}_k, \boldsymbol{Y}_k)$. In order to evaluate a closed form expression for \eqref{momentGenPartialUk}, we write
% \begin{align}
% &\E_{p_{W, X^n, Y^n}}\left[e^{t\sum_{k=1}^n \lambda(\boldsymbol{X}_k, \boldsymbol{Y}_k)}\right]  \notag\\
%&= \int_{w, x^{n}, y^{n}} p_{W, X^{n}, Y^{n}}(w, x^{n},y^{n})e^{t\sum_{k=1}^{n} \lambda(x_k,y_k)}
%  \, \mathrm{d}y^n \, \mathrm{d}x^n\, \mathrm{d}w \notag\\
%  & \stackrel{\eqref{powerConstraintCharacteristicFunction}}{=} \int_{w, x^{n}, y^{n}} p_{W, X^{n}, Y^{n}}(w, x^{n},y^{n}) \mathbf{1}\left(\left\{\sum_{k=1}^{n}x_k^2 = nP\right\}\right)  e^{t\sum_{k=1}^{n} \lambda(x_k,y_k)}
%  \, \mathrm{d}y^n \, \mathrm{d}x^n\, \mathrm{d}w \notag\\
%  & = \int\limits_{w, x^{n}, y^{n}} p_{W, X^{n}, Y^{n}}(w, x^{n},y^{n}) \mathbf{1}\left(\left\{\sum_{k=1}^{n}x_k^2 = nP\right\}\right)  e^{t\left(\sum\limits_{k=1}^{n} \lambda(x_k,y_k)\right)+\frac{2t^2}{1+2tP} \left(nP - \sum\limits_{k=1}^{n}x_k^2\right)}
%  \, \mathrm{d}y^n \, \mathrm{d}x^n\, \mathrm{d}w \notag\\
%   & \stackrel{\eqref{powerConstraintCharacteristicFunction}}{=} \int\limits_{w, x^{n}, y^{n}} p_{W, X^{n}, Y^{n}}(w, x^{n},y^{n}) e^{t\left(\sum\limits_{k=1}^{n} \lambda(x_k,y_k)\right)+\frac{2t^2}{1+2tP} \left(nP - \sum\limits_{k=1}^{n}x_k^2\right)}
%  \, \mathrm{d}y^n \, \mathrm{d}x^n\, \mathrm{d}w \notag\\
%  &  \stackrel{\text{(a)}}{=}\frac{1}{|\mathcal{W}|}\sum\limits_{w\in \mathcal{W}}\int_{x^{n}, y^{n}} p_{X^{n}, Y^{n}|W}(x^{n},y^{n}|w)e^{t\left(\sum\limits_{k=1}^{n} \lambda(x_k,y_k)\right)+\frac{2t^2}{1+2tP} \left(nP - \sum\limits_{k=1}^{n}x_k^2\right)}
%  \, \mathrm{d}y^n \, \mathrm{d}x^n %\label{eqnBHT4thChain}
%\end{align}
% where (a) follows from Definition~\ref{defCode} that $W$ is uniform on $\mathcal{W}$.
%% \begin{enumerate}
%% \item[(a)] follows from \eqref{powerConstraintInProofn} that
%%%\[
%%%\Pr_{p_{W,X^n, Y^n}}\left\{\sum_{k=1}^n  \boldsymbol{X}_k^2 = nP \right\}=1.
%%%\]
%%\begin{equation}
%%\frac{2t^2}{1+2tP} \left(nP - \sum\limits_{k=1}^{n}x_k^2\right)=0 \label{stepAeqnBHT4thChain}
%%\end{equation}
%%for all $x^n$ on the support of $p_{X^n}$.
% Following \eqref{eqnBHT4thChain}, consider the following chain of equalities for each $w\in \mathcal{W}$ and each $\ell\in \{0, 1, \ldots, n-2\}$:
% \begin{align}
% & \int\limits_{x^{n-\ell}, y^{n-\ell}} p(x^{n-\ell},y^{n-\ell}|w)e^{t\left(\sum\limits_{k=1}^{n-\ell} \lambda(x_k,y_k)\right) +\frac{2t^2}{1+2tP} \left(nP - \sum\limits_{k=1}^{n-\ell}x_k^2\right)}
%  \, \mathrm{d}y^{n-\ell} \, \mathrm{d}x^{n-\ell} \notag\\
% & = \int\limits_{x^{n-\ell-1}, y^{n-\ell-1}} p(x^{n-\ell-1},y^{n-\ell-1}|w)e^{t\left(\sum\limits_{k=1}^{n-\ell-1} \lambda(x_k,y_k)\right) +\frac{2t^2}{1+2tP} \left(nP - \sum\limits_{k=1}^{n-\ell-1}x_k^2\right)} \notag\\
% & \qquad \times
%   \int\limits_{x_{n-\ell}, y_{n-\ell}}   p(x_{n-\ell},y_{n-\ell}|w, x^{n-\ell-1}, y^{n-\ell-1}) e^{t\lambda(x_{n-\ell},y_{n-\ell}) -\frac{2t^2x_{n-\ell}^2}{1+2tP} }\, \mathrm{d}y_{n-\ell} \, \mathrm{d}x_{n-\ell} \, \mathrm{d}y^{n-\ell-1} \, \mathrm{d}x^{n-\ell-1}\notag\\
%   & \stackrel{\text{(a)}}{=} \int\limits_{x^{n-\ell-1}, y^{n-\ell-1}} p(x^{n-\ell-1},y^{n-\ell-1}|w)e^{t\left(\sum\limits_{k=1}^{n-\ell-1} \lambda(x_k,y_k)\right) +\frac{2t^2}{1+2tP} \left(nP - \sum\limits_{k=1}^{n-\ell-1}x_k^2\right)} \notag\\
% & \qquad \times
%   \int\limits_{x_{n-\ell}}   p(x_{n-\ell}|w,x^{n-\ell-1}, y^{n-\ell-1}) e^{\frac{-2t^2 x_{n-\ell}^2}{1+2tP}} \notag\\
%    &\qquad \times \int\limits_{y_{n-\ell}} \phi_{0,1}(y_{n-\ell}-x_{n-\ell}) e^{t \lambda(x_{n-\ell},y_{n-\ell})}\, \mathrm{d}y_{n-\ell} \, \mathrm{d}x_{n-\ell} \, \mathrm{d}y^{n-\ell-1} \, \mathrm{d}x^{n-\ell-1}\notag\\
%   & \stackrel{\text{(b)}}{=} \int\limits_{x^{n-\ell-1}, y^{n-\ell-1}} p(x^{n-\ell-1},y^{n-\ell-1}|w)e^{t\left(\sum\limits_{k=1}^{n-\ell-1} \lambda(x_k,y_k)\right) +\frac{2t^2}{1+2tP} \left(nP - \sum\limits_{k=1}^{n-\ell-1}x_k^2\right)} \notag\\
% & \qquad \times
%   \int\limits_{x_{n-\ell}}   p(x_{n-\ell}|w,x^{n-\ell-1}, y^{n-\ell-1}) e^{\frac{-2t^2 x_{n-\ell}^2}{1+2tP}} \notag\\
%    &\qquad \times \frac{1}{\sqrt{1+2tP}}e^{\frac{2t^2x_{n-\ell}^2}{1+2tP}} \, \mathrm{d}x_{n-\ell} \, \mathrm{d}y^{n-\ell-1} \, \mathrm{d}x^{n-\ell-1}\notag\\
%   &   = \frac{1}{\sqrt{1+2tP}}\int\limits_{x^{n-\ell-1}, y^{n-\ell-1}} p(x^{n-\ell-1},y^{n-\ell-1}|w)e^{t\left(\sum\limits_{k=1}^{n-\ell-1} \lambda(x_k,y_k)\right) +\frac{2t^2}{1+2tP} \left(nP - \sum\limits_{k=1}^{n-\ell-1}x_k^2\right)} \, \mathrm{d}y^{n-\ell-1} \, \mathrm{d}x^{n-\ell-1} \label{reverseInductionStep1}
% \end{align}
% where
% \begin{enumerate}
% \item[(a)] follows from \eqref{memorylessStatement*} and \eqref{memorylessStatement++}.
% \item[(b)] follows from evaluating the integral
%\begin{align*}
%& \int\limits_{y_{n-\ell}} \phi_{0,1}(y_{n-\ell}-x_{n-\ell}) e^{t \lambda(x_{n-\ell},y_{n-\ell})}\, \mathrm{d}y_{n-\ell} \\*
% & = \frac{1}{\sqrt{2\pi}} \int\limits_{y_{n-\ell}}   e^{- (y_{n-\ell}-x_{n-\ell})^2/2}e^{t(-P(y_{n-\ell}-x_{n-\ell})^2+2 x_{n-\ell}(y_{n-\ell}-x_{n-\ell}) ) }\,   \mathrm{d}y_{n-\ell} \\
% &=\int\limits_{z}\frac{1}{\sqrt{2\pi}} e^{-z^2/2}e^{t ( -Pz^2 + 2 x_{n-\ell} z)}\, \mathrm{d}z \\
% & =\frac{1}{\sqrt{2\pi}} \sqrt{ \frac{\pi}{\frac{1}{2} + tP}}e^{ \frac{(2tx_{n-\ell})^2 }{4 \left(\frac{1}{2}+tP\right)}} \\
%&= \frac{1}{\sqrt{1+2tP}}e^{\frac{2t^2x_{n-\ell}^2}{1+2tP}}
%\end{align*}
% by using the definition of $\lambda(\cdot,\cdot)$ in  \eqref{defLambda} and the substitution
% \[
% \phi_{0,1}(y_{n-\ell}-x_{n-\ell})=\frac{1}{\sqrt{2\pi}}e^{-(y_{n-\ell}-x_{n-\ell})^2/2}.
% \]
% \end{enumerate}
%Applying \eqref{reverseInductionStep1} recursively from $\ell=0$ to $\ell=n-2$, we have for each $w\in\mathcal{W}$
%\begin{align}
%& \int_{x^{n}, y^{n}} p_{X^{n}, Y^{n}|W}(x^{n},y^{n}|w)e^{t(\sum_{k=1}^{n} \lambda(x_k,y_k)) +\frac{2t^2}{1+2tP} (nP - \sum_{k=1}^{n}x_k^2)}
%  \, \mathrm{d}y^n \, \mathrm{d}x^n \notag\\
%  & = {(1+2tP)}^{-\frac{1}{2}}\int_{x^{n-1}, y^{n-1}} p_{X^{n-1}, Y^{n-1}|W}(x^{n-1},y^{n-1}|w)e^{t(\sum_{k=1}^{n-1} \lambda(x_k,y_k)) +\frac{2t^2}{1+2tP} (n P - \sum_{k=1}^{n-1}x_k^2)}
%  \, \mathrm{d}y^{n-1} \, \mathrm{d}x^{n-1} \notag\\
%  & ={(1+2tP)}^{-1}\int_{x^{n-2}, y^{n-2}} p_{X^{n-2}, Y^{n-2}|W}(x^{n-2},y^{n-2}|w)e^{t(\sum_{k=1}^{n-2} \lambda(x_k,y_k)) +\frac{2t^2}{1+2tP} ( nP - \sum_{k=1}^{n-2}x_k^2)}
%  \, \mathrm{d}y^{n-2} \, \mathrm{d}x^{n-2} \notag\\
%  &\:\:\: \vdots\notag\\
% & = {(1+2tP)}^{-\frac{n-1}{2}} \int_{x_1, y_1} p_{X_1, Y_1|W}(x_1,y_1|w) e^{t\lambda(x_1,y_1) +\frac{2t^2}{1+2tP} (nP - x_1^2)}
%  \, \mathrm{d}y_1 \, \mathrm{d}x_1,
%\label{reverseInductionStep2}
%\end{align}
%where the $k^{\text{th}}$ equality follows from \eqref{reverseInductionStep1} for $\ell=k-1$.
%Following \eqref{reverseInductionStep2}, we consider the following chain of equalities for each $w\in \mathcal{W}$:
%\begin{align*}
%&\int_{x_1, y_1} p_{X_1, Y_1|W}(x_1,y_1|w) e^{t\lambda(x_1,y_1) +\frac{2t^2}{1+2tP} (nP - x_1^2)}
%  \, \mathrm{d}y_1 \, \mathrm{d}x_1 \notag\\
%  & \stackrel{\eqref{memorylessStatement*}}{=} \int_{x_1} p_{X_1|W}(x_1|w) e^{\frac{2t^2}{1+2tP} (nP - x_1^2)} \int_{y_1} p_{Y_1|X_1}(y_1|x_1) e^{t\lambda(x_1,y_1)}
%  \, \mathrm{d}y_1 \, \mathrm{d}x_1 \notag\\
% &  \stackrel{\eqref{memorylessStatement++}}{=}  \int_{x_1} p_{X_1|W}(x_1|w) e^{\frac{2t^2}{1+2tP} (nP - x_1^2)}  \frac{1}{\sqrt{(1+2tP)}} e^{\frac{2t^2x_1^2}{1+2tP}}
%  \, \mathrm{d}x_1 \notag\\
%  & =  \frac{1}{\sqrt{(1+2tP)}} e^{\frac{2t^2nP}{1+2tP}},
%\end{align*}
%which implies from \eqref{reverseInductionStep2} that
%\begin{equation}
% \int_{x^{n}, y^{n}} p_{X^{n}, Y^{n}|W}(x^{n},y^{n}|w)e^{t\sum_{k=1}^{n} \lambda(x_k,y_k)}
%  \, \mathrm{d}y^n \, \mathrm{d}x^n  = {(1+2tP)}^{-\frac{n}{2}} e^{\frac{2t^2nP}{1+2tP}}.\label{reverseInductionStep3}
%\end{equation}
%Combining \eqref{eqnBHT4thChain} and \eqref{reverseInductionStep3}, we have
%\begin{equation}
%\E_{p_{W,X^n, Y^n}}\left[e^{t\sum_{k=1}^n \lambda(\boldsymbol{X}_k,\boldsymbol{Y}_k)}\right] = {(1+2tP)}^{-\frac{n}{2}} e^{\frac{2t^2nP}{1+2tP}}. \label{momentGenSumXkZk}
%\end{equation}
%Let $\{\boldsymbol{Z}_k\}_{k=1}^n$ be $n$ independent copies of the standard Gaussian random variable. A straightforward calculation reveals that
%\begin{equation}
%\E_{\prod_{k=1}^n p_{\boldsymbol{Z}_k}}\left[e^{t\sum_{k=1}^n (-P\boldsymbol{Z}_k^2 + 2\sqrt{P}\boldsymbol{Z}_k)}\right] = {(1+2tP)}^{-\frac{n}{2}} e^{\frac{2t^2nP}{1+2tP}}. \label{momentGenSumXkZk*}
%\end{equation}
%Therefore,
%\begin{equation}
%\E_{p_{W,X^n, Y^n}}\left[e^{t\sum_{k=1}^n \lambda(\boldsymbol{X}_k,\boldsymbol{Y}_k)}\right] = \E_{\prod_{k=1}^n p_{\boldsymbol{Z}_k}}\left[e^{t\sum_{k=1}^n (-P\boldsymbol{Z}_k^2 + 2\sqrt{P}\boldsymbol{Z}_k)}\right] \label{momentGenSumXkZk**}
%\end{equation}
% by \eqref{momentGenSumXkZk} and \eqref{momentGenSumXkZk*}, i.e., the moment generating functions of $\sum_{k=1}^n (-P\boldsymbol{Z}_k^2 + 2\sqrt{P}\boldsymbol{Z}_k)$ and $\sum_{k=1}^n \lambda(\boldsymbol{X}_k,\boldsymbol{Y}_k)$ are equal. \vspace{0.04 in} \linebreak It then follows that the probability distributions of
%$\sum_{k=1}^n (-P\boldsymbol{Z}_k^2 + 2\sqrt{P}\boldsymbol{Z}_k)$ and $\sum_{k=1}^n \lambda(\boldsymbol{X}_k,\boldsymbol{Y}_k)$ are equal, which \vspace{0.04 in} \linebreak implies from \eqref{defUk} and \eqref{defLambda} that the probability distributions of $\sum_{k=1}^n\frac{}{2(1+P)}(-P\boldsymbol{Z}_k^2 + 2\sqrt{P}\boldsymbol{Z}_k+P)$ and $\sum_{k=1}^n U_k$ are equal, which then implies from \eqref{eqnBHT6thChain} that
%\begin{align}
%&\log M_{\mathrm{fb}}^*(n-1, \varepsilon, P) \notag\\
%&\le \log\xi_n - \log\Bigg(\Pr_{\prod_{k=1}^n p_{\boldsymbol{Z}_k}} \left\{\sum_{k=1}^n\frac{}{2(1+P)}(-P\boldsymbol{Z}_k^2 + 2\sqrt{P}\boldsymbol{Z}_k+P) < \log\xi_n - \frac{n}{2}\log(1+P) \right\}-\varepsilon \Bigg). \label{eqnBHT7thChain}
%\end{align}
%\subsection{Applying the  Berry-Ess\'een Theorem}
%%Although the remaining steps for simplifying \eqref{eqnBHT7thChain} are standard (cf.\ \cite[Th.~74]{Pol10} and \cite[Th.~4.4]{TanBook}), we include them for completeness. We define the mean of the random variable in \eqref{eqnBHT7thChain} as
%%\begin{align*}
%%\mu &\stackrel{\text{def}}{=} \E_{p_{Z_1}}\left[\frac{}{2(1+P)}(-PZ_1^2 + 2\sqrt{P}Z_1+P)\right] \\
%%& =0,
%%\end{align*}
%%the standard deviation as
%%\begin{align}
%%\sigma & \stackrel{\text{def}}{=} \sqrt{\Var_{p_{Z_1}}\left[\left(\frac{}{2(1+P)}(-PZ_1^2 + 2\sqrt{P}Z_1+P)\right)^2\right]} \notag\\
%%&=\sqrt{\frac{P(P+2)}{2(1+P)^2}} \label{defSigma}
%%\end{align}
%%and the third absolute moment as
%%\begin{equation}
%%T\stackrel{\text{def}}{=} \E_{p_{Z_1}}\left[\left|\frac{}{2(1+P)}(-PZ_1^2 + 2\sqrt{P}Z_1+P)\right|^3\right]. \label{defT}
%%%& = \left(\frac{}{2(1+P)}\right)^3(15(P+1)^3  + 18P(P+1)^2 + 12P^2(P+1)+ 8P^3),
%%\end{equation}
%%Since
%%\begin{align*}
%%T^{1/3}& \stackrel{\eqref{defT}}{=} \left(\E_{p_{Z_1}}\left[\left|\frac{}{2(1+P)}(-PZ_1^2 + 2\sqrt{P}Z_1+P)\right|^3\right]\right)^{1/3}\notag\\
%%& \stackrel{\text{(a)}}{\le}  \frac{}{2(1+P)}\left(P\left(\E_{p_{Z_1}}\left[Z_1^6\right]\right)^{1/3}+2\sqrt{P}\left(\E_{p_{Z_1}}\left[|Z_1|^3\right]\right)^{1/3}+P\right)\notag\\
%%& = \frac{}{2(1+P)}\left(15^{1/3}P+2(2\sqrt{2/\pi})^{1/3} \sqrt{P}+P\right)
%%\end{align*}
%%where (a) follows from the triangle inequality for the $3$-norm, it follows that $T$ is finite.
%%Using \eqref{defSigma} and \eqref{defT} and applying Berry-Ess\'een theorem for i.i.d.\ random variables \cite[Section XVI.5]{feller}, we have the following bound for all $n\in\mathbb{N}$:
%%\begin{align*}
%%\sup_{a\in\mathbb{R}}\left|\Pr_{\prod_{k=1}^n p_{\boldsymbol{Z}_k}}\left\{\frac{1}{\sigma\sqrt{n}}\sum_{k=1}^n  \frac{}{2(1+P)}( -P\boldsymbol{Z}_k^2 + 2\sqrt{P}\boldsymbol{Z}_k+P) \le a\right\} - \Phi(a) \right| \le \frac{T}{\sigma^3 \sqrt{n}}.
%%\end{align*}
%%This implies by choosing $a= \Phi^{-1}\left(\varepsilon +\frac{2T}{\sigma^3 \sqrt{n}}\right)$ that
%%\begin{align}
%%\Pr_{\prod_{k=1}^n p_{\boldsymbol{Z}_k}}\left\{\frac{1}{\sigma\sqrt{n}}\sum_{k=1}^n  \frac{}{2(1+P)}( -P\boldsymbol{Z}_k^2 + 2\sqrt{P}\boldsymbol{Z}_k+P) <\Phi^{-1}\left(\varepsilon + \frac{2T}{\sigma^3 \sqrt{n}}\right)\right\} > \varepsilon + \frac{T}{\sigma^3 \sqrt{n}}\,. \label{BerryEsseenSt1}
%%\end{align}
%%Following \eqref{eqnBHT7thChain} and letting
%%\begin{equation*}
%%\xi_n \stackrel{\text{def}}{=} \frac{n}{2}\log(1+P) + \sigma\sqrt{n}\Phi^{-1}\left(\varepsilon + \frac{2T}{\sigma^3 \sqrt{n}}\right),
%%\end{equation*}
%%we can express \eqref{eqnBHT7thChain} as
%%\begin{align*}
%%&\log M_{\mathrm{fb}}^*(n-1, \varepsilon, P) \notag\\
%%&\le \frac{n}{2}\log(1+P) + \sigma\sqrt{n}\Phi^{-1}\left(\varepsilon + \frac{2T}{\sigma^3 \sqrt{n}}\right) \notag\\
%% & \qquad - \log\Bigg(\Pr_{\prod_{k=1}^n p_{\boldsymbol{Z}_k}} \left\{\sum_{k=1}^n\frac{}{2(1+P)}(-P\boldsymbol{Z}_k^2 + 2\sqrt{P}\boldsymbol{Z}_k+P) <\sigma\sqrt{n}\Phi^{-1}\left(\varepsilon + \frac{2T}{\sigma^3 \sqrt{n}}\right)  \right\}-\varepsilon \Bigg),
%%\end{align*}
%%which implies from \eqref{BerryEsseenSt1} that
%%\begin{align}
%%\log M_{\mathrm{fb}}^*(n-1, \varepsilon, P) < \frac{n}{2}\log(1+P) + \sigma\sqrt{n}\Phi^{-1}\left(\varepsilon + \frac{2T}{\sigma^3 \sqrt{n}}\right) - \log \frac{T}{\sigma^3 \sqrt{n}}\,. \label{eqnBHT8thChain}
%%\end{align}
%%Since
%%\[
%%\Phi^{-1}\left(\varepsilon + \frac{2T}{\sigma^3 \sqrt{n}}\right) = \Phi^{-1}(\varepsilon) + \frac{2T}{\sigma^3 \sqrt{n}} \left(\Phi^{-1}\right)^\prime(c)
%%\]
%%for some $c\in [\varepsilon, 2T/\sigma^3]$ by Taylor's theorem, it follows from \eqref{eqnBHT8thChain} that there exists some real constant
%%\begin{equation}
%%\bar \kappa \stackrel{\text{def}}{=}  \frac{2T}{\sigma^2}\left(\Phi^{-1}\right)^\prime(c) -\log \frac{T}{\sigma^3} \label{defBarKappa}
%%\end{equation}
%% that does not depend on~$n$ (cf.\ \eqref{defSigma} and \eqref{defT}) such that
%%\begin{equation*}
%%\log M_{\mathrm{fb}}^*(n-1, \varepsilon, P) < \frac{n}{2}\log(1+P) + \sigma\sqrt{n}\Phi^{-1}(\varepsilon) + \frac{1}{2}\log n + \bar \kappa\,,
%%\end{equation*}
%%which implies by letting
%%\begin{equation}
%%\kappa\stackrel{\text{def}}{=} \bar\kappa + \frac{1}{2}\log(1+P) + \sigma\Phi^{-1}(\varepsilon) + \frac{1}{2} \label{defKappa}
%%\end{equation}
%%that
%%\begin{equation}
%%\log M_{\mathrm{fb}}^*(n-1, \varepsilon, P) < \frac{n-1}{2}\log(1+P) + \sigma\sqrt{n-1}\Phi^{-1}(\varepsilon) + \frac{1}{2}\log (n-1) + \kappa   \, \label{eqnBHT9thChain}
%%\end{equation}
%%for $n\ge 2$. Combining \eqref{defCPparallelNFB}, \eqref{defVP}, \eqref{defSigma}, \eqref{defT}, \eqref{defBarKappa} \eqref{defKappa} and \eqref{eqnBHT9thChain}, we have \eqref{cutsetStatement}.
%\section{Parallel Gaussian Channels with Feedback} \label{sectionParallelAWGNChannel}
%\subsection{Problem Setup and Main Result}
%
%
%\subsection{Strong Converse}
%It was shown in Tan and Tomamichel's work~\cite[Appendix~A]{TanTom13a} that for each $\varepsilon\in(0,1)$, there exists a constant $\hat \kappa$  (not depending on~$n$) such that
%\begin{equation}
%\log M_{\mathrm{fb}}^*(n, \varepsilon, P) \ge n \mathrm{C}(\mathbf{P}^*) + \sqrt{n\mathrm{V}(\mathbf{P}^*)}\Phi^{-1}(\varepsilon) +  \frac{1}{2}\log n + \hat \kappa, \label{cutsetStatementParallelAch}
%\end{equation}
%where $\mathrm{C}(\mathbf{P}^*)$ was defined in \eqref{defCPparallelNFB} and
%\begin{equation}
%\mathrm{V}(\mathbf{P}^*) \stackrel{\text{def}}{=} \sum_{\ell=1}^L \mathrm{V}(P_\ell/N_\ell) \qquad\mbox{bits$^2$ per channel use} \label{defVPparallel}
%\end{equation}
%denotes the dispersion of the parallel Gaussian channel without feedback (first proved by  Polyanskiy~\cite[Theorem~78]{Pol10}).
%%. The dispersion of this class of channels (without the $\frac{1}{2}\log n$ third-order term) was .
%Theorem~\ref{thmMainResultParallel} together with \eqref{cutsetStatementParallelAch} imply that
%\begin{equation}
%\lim_{n\to\infty}\frac{1}{n}\log M_{\mathrm{fb}}^*(n, \varepsilon, P)=\mathrm{C}(\mathbf{P}^*)
%\end{equation}
%for all $\varepsilon\in (0,1)$. Since the limit of the normalized logarithm of the code sizes exists and  does not depend on $\varepsilon \in (0,1)$, the strong converse is established for the parallel Gaussian channel with feedback.
%
%\subsection{Proof of Theorem \ref{thmMainResultParallel}}
%\begin{IEEEproof}
%Fix an $\varepsilon\in (0,1)$ and choose an arbitrary sequence of $(\bar n, M_{\mathrm{fb}}^*(\bar n, \varepsilon, P), P, \varepsilon)$-feedback codes for the parallel Gaussian channel with feedback. Letting $n=\bar n+1$ and following similar procedures for proving \eqref{eqnBHTSecondChain} at the start of the proof of Theorem~\ref{thmMainResult}, we obtain a sequence of $(n, M_{\mathrm{fb}}^*(n-1, \varepsilon, P), P, \varepsilon)$-feedback codes with
% \begin{equation}
% \Pr\left\{\sum_{\ell=1}^L \sum_{k=1}^n X_{\ell,k}^2 = n P\right\}=1 \label{powerEqualityParallel}
% \end{equation}
% such that the following inequality holds for each $n\in \{2, 3, \ldots\}$ and each $\xi_n >0$:
%\begin{align}
%&\log M_{\mathrm{fb}}^*(n-1, \varepsilon, P)\notag\\
%& \le \log\xi_n - \log\left(\Pr_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}} \left\{\sum_{k=1}^n \log\frac{q_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{Y}_{k}|\boldsymbol{X}_{k})}{r_{\boldsymbol{Y}_{k}}(\boldsymbol{Y}_{k})} < \log\xi_n \right\}-\varepsilon\right),\label{eqnBHTSecondChainParallel}
%\end{align}
%where $p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}$ denotes the probability distribution induced by the $(n, M_{\mathrm{fb}}^*(n-1, \varepsilon, P), P, \varepsilon)$-feedback code, and $r_{\boldsymbol{Y}_{k}}$ is defined for each $k\in\{1, 2, \ldots, n\}$ as
%\begin{equation}
%r_{\boldsymbol{Y}_{k}}(\mathbf{y}_{k})=\prod_{\ell=1}^L \phi_{0, P_\ell + N_\ell}(y_{\ell,k}) \label{defsYkParallel}
%\end{equation}
%for all $\mathbf{y}_{k}$. The channel law is
%\begin{equation}
%q_{\boldsymbol{Y}|\boldsymbol{X}}(\mathbf{y}_k|\mathbf{x}_k)=\prod_{\ell=1}^L \phi_{0,N_\ell}(y_{\ell,k}-x_{\ell,k}) \label{memorylessStatement++parallel}
%\end{equation}
%for each $k\in\{1, 2, \ldots, n\}$. Combining \eqref{defsYkParallel} and \eqref{memorylessStatement++parallel}, we have
%\begin{align}
%& \log\frac{q_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{Y}_k|\boldsymbol{X}_k)}{r_{\boldsymbol{Y}_k}(\boldsymbol{Y}_k)} \notag\\
%& = \sum_{\ell=1}^L \left( \frac{1}{2}\log\left(1+\frac{P_\ell}{N_\ell}\right)+ \frac{}{2(N_\ell+P_\ell)}\left(\frac{-P_\ell}{N_\ell}(Y_{\ell,k}-X_{\ell,k})^2 + X_{\ell,k}^2 + 2X_{\ell,k}(Y_{\ell,k}-X_{\ell,k})\right)\right)\notag\\
%&\stackrel{\text{(a)}}{=}\sum_{\ell=1}^L \left( \frac{1}{2}\log\left(1+\frac{P_\ell}{N_\ell}\right)+ \frac{}{2\Lambda}\left(\frac{-P_\ell}{N_\ell}(Y_{\ell,k}-X_{\ell,k})^2 + X_{\ell,k}^2 + 2X_{\ell,k}(Y_{\ell,k}-X_{\ell,k})\right)\right) \label{defLogLikelihoodParallel}
%\end{align}
%for each $k\in\{1, 2, \ldots, n\}$, where (a) follows from \eqref{condition(ii)}. Following \eqref{defLogLikelihoodParallel}, we define the function $\lambda: \mathbb{R}^4 \rightarrow \mathbb{R}$ such that
%\begin{equation}
%\lambda(P, \sigma^2, x,y)\stackrel{\text{def}}{=} \frac{-P}{\sigma^2}(y-x)^2 + P + 2x(y-x) \label{defLambdaEllParallel}
%\end{equation}
%and let
%\begin{equation}
%U_{k} \stackrel{\text{def}}{=} \sum_{\ell=1}^L \lambda(P_\ell, N_\ell, X_{\ell,k},Y_{\ell,k}) \label{defUkParallel}
%\end{equation}
%for each $k\in\{1, 2, \ldots, n\}$. It then follows from \eqref{powerEqualityParallel}, \eqref{defLambdaEllParallel} and \eqref{defUkParallel} that
%\begin{equation*}
%\Pr_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}}\left\{\sum_{k=1}^n U_{k}= \sum_{k=1}^n \sum_{\ell=1}^L  \left(\frac{-P_\ell}{N_\ell}(Y_{\ell,k}-X_{\ell,k})^2 + X_{\ell,k}^2 + 2X_{\ell,k}(Y_{\ell,k}-X_{\ell,k})\right)\right\}=1,
%\end{equation*}
%which implies from \eqref{defCPparallelNFB}, \eqref{eqnBHTSecondChainParallel} and \eqref{defLogLikelihoodParallel} that
%\begin{align}
%\log M_{\mathrm{fb}}^*(n-1, \varepsilon, P)\le \log\xi_n - \log\left(\Pr_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}} \left\{\sum_{k=1}^n U_{k} < \frac{2\Lambda(\log\xi_n - n \mathrm{C}(\mathbf{P}^*))}{} \right\}-\varepsilon\right).\label{eqnBHT3rdChainParallel}
%\end{align}
%In the rest of the proof, we would like to use Chebyshev's inequality to bound the probability term in \eqref{eqnBHT3rdChainParallel}. To this end, we will evaluate in the following
%\begin{equation}
%\E_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}}\left[ \sum_{k=1}^n U_k \right]\stackrel{\eqref{defUkParallel}}{=}\E_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}}\left[ \sum_{k=1}^n \sum_{\ell=1}^L \lambda(P_\ell, N_\ell, X_{\ell,k},Y_{\ell,k}) \right] \label{evaluateExpSumUk1st}
% \end{equation}
% and
% \begin{align}
% &\Var_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}}\left[\sum_{k=1}^n U_k \right] \notag\\*
% & \stackrel{\eqref{defUkParallel}}{=}\Var_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}}\left[ \sum_{k=1}^n \sum_{\ell=1}^L \lambda(P_\ell, N_\ell, X_{\ell,k},Y_{\ell,k}) \right]\notag\\
% & = \sum_{k=1}^n \sum_{m=1}^n \sum_{\ell=1}^L \sum_{\ell^\prime =1}^L \Cov\left[  \lambda(P_\ell, N_\ell, X_{\ell,k},Y_{\ell,k}), \lambda(P_{\ell^\prime}, N_{\ell^\prime}^2, x_{\ell^\prime,m},y_{\ell^\prime,m}) \right] \notag\\
% & = \sum_{k=1}^n \sum_{\ell=1}^L \Var_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}}\left[ \lambda(P_\ell, N_\ell, X_{\ell,k},Y_{\ell,k}) \right] + \sum_{(m, \ell^\prime)\ne (k, \ell)}\Cov\left[  \lambda(P_\ell, N_\ell, X_{\ell,k},Y_{\ell,k}), \lambda(P_{\ell^\prime}, N_{\ell^\prime}^2, x_{\ell^\prime,m},y_{\ell^\prime,m}) \right].   \label{evaluateVarSumUk1st}
%  \end{align}
%Following \eqref{evaluateExpSumUk1st}, we consider the following chain of equalities for each $k\in\{1, 2, \ldots, n\}$ and each $\ell\in \mathcal{L}$:
%\begin{align}
%& \E_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}}\left[\lambda(P_\ell,N_\ell,X_{\ell,k},Y_{\ell,k})\right] \notag\\
%& \stackrel{\eqref{defLambdaEllParallel}}{=}  \E_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}}\left[\frac{-P_\ell}{N_\ell}(Y_{\ell,k}-X_{\ell,k})^2 + P_\ell + 2X_{\ell,k}(Y_{\ell,k}-X_{\ell,k})\right] \notag\\
%& \stackrel{\text{(a)}}{=} \E_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}}\left[ 2X_{\ell,k}(Y_{\ell,k}-X_{\ell,k})\right] \notag\\
%& \stackrel{\text{(b)}}{=} 2  \E_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}}\left[ X_{\ell,k}\right]\E_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}}\left[Y_{\ell,k}-X_{\ell,k}\right] \notag\\
%& \stackrel{\text{(c)}}{=}0, \label{evaluateExpSumUk2nd}
%\end{align}
%where
%\begin{enumerate}
%\item[(a)] follow from Statement (ii) in Proposition~\ref{propositionIndepNoises}.
%%\eqref{memorylessStatement++parallel} that $(Y_{\ell,k}-X_{\ell,k})$ is a Gaussian random variable whose variance is $N_\ell$.
%\item[(b)] follows from Statement (i) in Proposition~\ref{propositionIndepNoises}.
%%the fact that $X_{\ell,k}$ and $(Y_{\ell,k}-X_{\ell,k})$ are independent.
%\item[(c)] follows from Statement (ii) in Proposition~\ref{propositionIndepNoises}.
%%\eqref{memorylessStatement++parallel} that $(Y_{\ell,k}-X_{\ell,k})$ is a zero-mean Gaussian random variable.
%\end{enumerate}
%Combining \eqref{evaluateExpSumUk1st} and \eqref{evaluateExpSumUk2nd}, we have
% \begin{equation}
%\E_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}}\left[ \sum_{k=1}^n U_k \right] = 0. \label{expectationUkParallel}
%\end{equation}
%In addition, following \eqref{evaluateVarSumUk1st},
% we consider the following chain of equalities for each $\ell$, $\ell^\prime$, $k$ and $m$ such that $(\ell^\prime, m) \ne (\ell, k)$:
%\begin{align}
%& \E_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}}\Bigg[\lambda(P_\ell,N_\ell,X_{\ell,k} ,Y_{\ell,k}) \lambda(P_{\ell^\prime},N_{\ell^\prime}^2,X_{\ell^\prime,m} ,Y_{\ell^\prime,m}) \Bigg] \notag\\
%& \stackrel{\eqref{defLambdaEllParallel}}{=} \E_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}}\Bigg[\Bigg(\frac{-P_\ell}{N_\ell}(Y_{\ell,k}-X_{\ell,k})^2 + P_\ell + 2X_{\ell,k}(Y_{\ell,k}-X_{\ell,k})\Bigg) \notag\\
% &\qquad \qquad \times \Bigg(\frac{-P_{\ell^\prime}}{N_{\ell^\prime}^2}(Y_{\ell^\prime,m}-X_{\ell^\prime,m})^2 + P_{\ell^\prime} + 2X_{\ell^\prime,m}(Y_{\ell^\prime,m}-X_{\ell^\prime,m})\Bigg)\Bigg] \notag\\
% & \stackrel{\text{(a)}}{=}   \E_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}}\Bigg[2X_{\ell,k}(Y_{\ell,k}-X_{\ell,k})\left(\frac{-P_{\ell^\prime}}{N_{\ell^\prime}^2}(Y_{\ell^\prime,m}-X_{\ell^\prime,m})^2 + P_{\ell^\prime}\right) \notag\\
% &\qquad \qquad + 2X_{\ell^\prime,m}(Y_{\ell^\prime,m}-X_{\ell^\prime,m})\left(\frac{-P_\ell}{N_\ell}(Y_{\ell,k}-X_{\ell,k})^2 + P_\ell\right)\notag\\
% & \qquad \qquad + 4X_{\ell,k} X_{\ell^\prime,m}(Y_{\ell,k}-X_{\ell,k})(Y_{\ell^\prime,m}-X_{\ell^\prime,m})\Bigg] \notag\\
%  & \stackrel{\text{(b)}}{=} \begin{cases}  \E_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}}\left[2X_{\ell,k}(Y_{\ell,k}-X_{\ell,k})\right]\E_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}}\Bigg[\frac{-P_{\ell^\prime}}{N_{\ell^\prime}^2}(Y_{\ell^\prime,m}-X_{\ell^\prime,m})^2 + P_{\ell^\prime} \Bigg] & \text{if $k\le m$,} \vspace{0.04 in} \\  \E_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}}\left[2X_{\ell^\prime,m}(Y_{\ell^\prime,m}-X_{\ell^\prime,m})\right]\E_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}}\Bigg[\frac{-P_{\ell}}{N_{\ell}^2}(Y_{\ell,k}-X_{\ell,k})^2 + P_{\ell} \Bigg] & \text{if $k>m$} \end{cases} \notag\\
%   & \stackrel{\text{(c)}}{=} 0, \label{ExpectationLambdakLambdamParallel}
%\end{align}
%where
%\begin{enumerate}
%\item[(a)] follows from Proposition~\ref{propositionIndepNoises} that
%\[
%\E_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}}\left[\Bigg(\frac{-P_\ell}{N_\ell}(Y_{\ell,k}-X_{\ell,k})^2+P_\ell\Bigg)\Bigg(\frac{-P_{\ell^\prime}}{N_{\ell^\prime}^2}(Y_{\ell^\prime,m}-X_{\ell^\prime,m})^2+P_{\ell^\prime}\Bigg)\right]=0.
%\]
%%the fact if $(\ell^\prime, m) \ne (\ell, k)$, then $(Y_{\ell,k}-X_{\ell,k})$ and $(Y_{{\ell^\prime},m}-X_{{\ell^\prime},m})$ are two independent Gaussian random variables whose variances are $N_\ell$ and $N_{\ell^\prime}^2$ respectively.
%    \item[(b)] follows from Proposition~\ref{propositionIndepNoises} that:\\
%    (i) If $k\le m$, $(Y_{{\ell^\prime},m}-X_{{\ell^\prime},m})$ is a zero-mean random variable that is independent of $(X_{\ell^\prime,m},X_{\ell,k}, Y_{\ell,k})$.\\
%    (ii) If $k > m$, $(Y_{\ell,k}-X_{\ell,k})$ is a zero-mean random variable that is independent of $(X_{\ell,k} ,X_{\ell^\prime,m}, Y_{\ell^\prime,m})$.
%        \item[(c)] follows from Statement (ii) in Proposition~\ref{propositionIndepNoises}.
%        %the fact that $(Y_{\ell,k}-X_{\ell,k})$ and $(Y_{{\ell^\prime},m}-X_{{\ell^\prime},m})$ are two zero-mean random variables whose variances are  $N_{\ell}^2$ and $N_{\ell^\prime}^2$ respectively.
%\end{enumerate}
%Combining \eqref{evaluateExpSumUk2nd} and \eqref{ExpectationLambdakLambdamParallel}, we obtain
%\begin{align*}
%\Cov_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}}\Big[\lambda(P_\ell,N_\ell,X_{\ell,k} ,Y_{\ell,k}), \lambda(P_{\ell^\prime},N_{\ell^\prime}^2,X_{\ell^\prime,m} ,Y_{\ell^\prime,m}) \Big] =0
%\end{align*}
%for all $(\ell^\prime, m) \ne (\ell, k)$, which implies from \eqref{evaluateVarSumUk1st} and \eqref{evaluateExpSumUk2nd} that
%\begin{align}
% \Var_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}}\left[\sum_{k=1}^n U_k\right]=  \sum_{k=1}^n \sum_{\ell=1}^L \E_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}}\left[ \left(\lambda(P_\ell, N_\ell, X_{\ell,k},Y_{\ell,k})\right)^2 \right]. \label{varianceSumUkParallel}
%\end{align}
%Following \eqref{varianceSumUkParallel}, we consider the following chain of equalities for each $k\in\{1, 2, \ldots, n\}$ and each $\ell\in \mathcal{L}$:
%\begin{align}
%& \E_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}}\left[ \left(\lambda(P_\ell, N_\ell, X_{\ell,k},Y_{\ell,k})\right)^2 \right] \notag\\
%& \stackrel{\eqref{defLambdaEllParallel}}{=}  \E_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}}\Bigg[\Bigg(\frac{-P_\ell}{N_\ell}(Y_{\ell,k}-X_{\ell,k})^2 + P_\ell + 2X_{\ell,k}(Y_{\ell,k}-X_{\ell,k})\Bigg)^2\Bigg] \notag\\
% & \stackrel{\text{(a)}}{=} \E_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}}\Bigg[2P_\ell^2 + 4 N_\ellX_{\ell,k}^2\Bigg] \label{ExpectationUkUkParallel}
%\end{align}
%where (a) follows from Proposition~\ref{propositionIndepNoises} that $(Y_{\ell,k}-X_{\ell,k})/N_\ell$ is a standard Gaussian random variable independent of $X_{\ell,k}$.
%It then follows from \eqref{ExpectationUkUkParallel} and \eqref{powerEqualityParallel} that
%\begin{align}
%4nP \min_{\ell\in\mathcal{L}}\{N_\ell\} <\sum_{k=1}^n \sum_{\ell=1}^L \E_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}}\left[ \left(\lambda(P_\ell, N_\ell, X_{\ell,k},Y_{\ell,k})\right)^2\right] \le   2n \sum_{\ell=1}^L P_\ell^2 +  4nP\max_{\ell\in\mathcal{L}}\{N_\ell\}. \label{eqnBHT4thChainParallel}
%\end{align}
%Letting
%\begin{equation}
%\tilde \kappa \stackrel{\text{def}}{=} 4\min_{\ell\in\mathcal{L}}\{N_\ell\}P \label{defTildeKappaParallel}
%\end{equation}
%and
%\begin{equation}
%\bar \kappa \stackrel{\text{def}}{=} 2\sum_{\ell=1}^L P_\ell^2 + 4\max_{\ell\in\mathcal{L}}\{N_\ell\}P \label{defBarKappaParallel}
%\end{equation}
%be two positive real numbers, it follows from \eqref{varianceSumUkParallel}, \eqref{eqnBHT4thChainParallel}, \eqref{defTildeKappaParallel} and \eqref{defBarKappaParallel} that
%\begin{equation}
% n \tilde \kappa<\Var_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}}\left[\sum_{k=1}^n U_k\right] \le n \bar \kappa. \label{eqnBHT5thChainParallel}
%\end{equation}
%Omitting the distribution subscripts for probability, expectation and variance and letting
%\begin{equation}
%\log\xi_n \stackrel{\text{def}}{=} n\mathrm{C}(\mathbf{P}^*) + \frac{}{2\Lambda}\sqrt{\left(\frac{2}{1-\varepsilon}\right)\Var\left[\sum_{k=1}^n U_k\right]}\, ,
%\end{equation}
%it follows from \eqref{eqnBHT3rdChainParallel} that
%\begin{align}
%&\log M_{\mathrm{fb}}^*(n-1, \varepsilon, P)\notag\\
%&\le n\mathrm{C}(\mathbf{P}^*) + \frac{}{2\Lambda}\sqrt{\left(\frac{2}{1-\varepsilon}\right)\Var\left[\sum_{k=1}^n U_k\right]} - \log\left(1-\varepsilon - \Pr\left\{\sum_{k=1}^n U_{k} \ge \sqrt{\left(\frac{2}{1-\varepsilon}\right)\Var\left[\sum_{k=1}^n U_k\right]}\right\}\right). \label{eqnBHT6thChainParallel}
%\end{align}
%Since $\sqrt{\left(\frac{2}{1-\varepsilon}\right)\Var\left[\sum_{k=1}^n U_k\right]}>0$ by \eqref{eqnBHT5thChainParallel}, it follows from Chebyshev's inequality that
%\begin{align*}
% \Pr\left\{\sum_{k=1}^n U_{k} \ge \sqrt{\left(\frac{2}{1-\varepsilon}\right)\Var\left[\sum_{k=1}^n U_k\right]}\right\} \le (1-\varepsilon)/2,
%\end{align*}
%which implies from \eqref{eqnBHT6thChainParallel} that
%\begin{align}
%\log M_{\mathrm{fb}}^*(n-1, \varepsilon, P ,L)\le n\mathrm{C}(\mathbf{P}^*) + \frac{}{2\Lambda}\sqrt{\left(\frac{2}{1-\varepsilon}\right)\Var\left[\sum_{k=1}^n U_k\right]} - \log\left(\frac{1-\varepsilon}{2}\right). \label{eqnBHT7thChainParallel}
%\end{align}
%Define
%\begin{equation}
%\kappa\stackrel{\text{def}}{=} \frac{}{\Lambda}\sqrt{\frac{2\bar \kappa}{1-\varepsilon}} - \log\left(\frac{1-\varepsilon}{2}\right)+ \mathrm{C}(\mathbf{P}^*) \label{defKappaParallel}
%\end{equation}
%and continue the inequality in \eqref{eqnBHT7thChainParallel} for $n\ge 2$ as follows:
%\begin{align}
%&n\mathrm{C}(\mathbf{P}^*) + \frac{}{2\Lambda}\sqrt{\left(\frac{2}{1-\varepsilon}\right)\Var\left[\sum_{k=1}^n U_k\right]} - \log\left(\frac{1-\varepsilon}{2}\right) \notag\\
%&\stackrel{\eqref{eqnBHT5thChainParallel}}{\le}  n\mathrm{C}(\mathbf{P}^*) + \frac{}{2\Lambda}\sqrt{\frac{2\bar \kappa n}{1-\varepsilon}  } - \log\left(\frac{1-\varepsilon}{2}\right) \notag\\
%& \stackrel{\text{(a)}}{\le} n\mathrm{C}(\mathbf{P}^*) + (\sqrt{n-1}+1)\frac{}{2\Lambda}\sqrt{\frac{2\bar \kappa}{1-\varepsilon} } - \log\left(\frac{1-\varepsilon}{2}\right) \notag\\
%&\le (n-1)\mathrm{C}(\mathbf{P}^*) + \sqrt{n-1}\left(\frac{}{\Lambda}\sqrt{\frac{2\bar \kappa}{1-\varepsilon}} - \log\left(\frac{1-\varepsilon}{2}\right) + \mathrm{C}(\mathbf{P}^*)\right) \notag\\
%& \stackrel{\eqref{defKappaParallel}}{=} (n-1)\mathrm{C}(\mathbf{P}^*) + \kappa\sqrt{n-1}, \label{eqnBHT8thChainParallel}
%\end{align}
%where
%\begin{enumerate}
%\item[(a)] follows from the fact that $\sqrt{n} \le \sqrt{n-1}+1$.
%\item[(b)] follows from our assumption $n\ge 2$ that $1\le \sqrt{n-1}$.
%\end{enumerate}
%The theorem then follows from combining \eqref{eqnBHT7thChainParallel} and \eqref{eqnBHT8thChainParallel}.
%\end{IEEEproof}
%\medskip
%
%\subsection{Difficulties in Establishing the Exact Second-Order Asymptotics}
%Unlike the case for $L=1$ where we are able to provide a converse proof for \eqref{eqn:asymp_expans_fb}, we fail to obtain a matching converse  statement to  \eqref{cutsetStatementParallelAch} for $L>1$. Instead, we can only conclude from Theorem~\ref{thmMainResultParallel} and \eqref{cutsetStatementParallelAch} that the second-order asymptotics in the asymptotic expansion of $\log M_{\mathrm{fb}}^*(n, \varepsilon, P)$ increases at a rate no faster than $\sqrt{n}$ (which is good enough for the purpose of the strong converse). The difficulty in obtaining a matching converse statement to \eqref{cutsetStatementParallelAch} for $L>1$ can be roughly explained as follows: For $L=1$, we can always assume without loss of generality that $N_1=1$ and $P=P_1$, and the key equation to proving the reverse statement of \eqref{cutsetStatementParallelAch} is \eqref{powerConstraintCharacteristicFunction}, which enables the insertion of
%\begin{equation}
%e^{\frac{2t^2}{1+2tP} \left(nP - \sum\limits_{k=1}^{n}x_k^2\right)} \label{sumXk2=0whenL=1}
%\end{equation}
%in the third equality of \eqref{eqnBHT4thChain} and the cancellation of $e^{\frac{2t^2x_{n-\ell}^2}{1+2tP}}$ in the last step of \eqref{reverseInductionStep1}. Unfortunately for $L>1$, to prove the converse statement to~\eqref{cutsetStatementParallelAch}, it appears to be necessary to ensure that the following is true:
%%the $e^{\frac{2t^2\sigma^2}{1+2tP} \left(nP - \sum\limits_{k=1}^{n}x_k^2\right)}$
%%if we want to imitate the insertion of $e^{\frac{2t^2\sigma^2}{1+2tP} \left(nP - \sum\limits_{k=1}^{n}x_k^2\right)}$ in order to prove the converse statement of \eqref{cutsetStatementParallelAch}, then we need to ensure
%\begin{equation}
%e^{\sum\limits_{\ell=1}^L\frac{2t^2 N_\ell}{1+2tP_\ell}\left(nP_\ell - \sum\limits_{k=1}^n x_{\ell,k}^2\right)}=1 . \label{sumXk2=0whenL>1}
%\end{equation}
%This requires the following $L$ equations to hold
% \begin{equation}
%p_{W,X_\ell^n, Y_\ell^n}(w,x_\ell^n, y_\ell^n) = p_{W,X_\ell^n, Y_\ell^n}(w,x_\ell^n, y_\ell^n)\mathbf{1}\left(\left\{\sum_{k=1}^{n}x_{\ell,k}^2 = nP_\ell\right\}\right),\qquad\forall\, \ell\in \mathcal{L}.\label{powerConstraintCharacteristicFunctionParallel}
% \end{equation}
%%for all $w\in\mathcal{W}$, $x_\ell^n\in \mathbb{R}^n$ and $y_\ell^n\in\mathbb{R}^n$.
%%
%%for all $x^n \in\mathbb{R}^n$ on the support of $p_{X^n}$, which can no longer be derived from \eqref{powerEqualityParallel}
%
%Unfortunately, we cannot assume (without loss of generality) that~\eqref{powerConstraintCharacteristicFunctionParallel} is true in view of \eqref{powerEqualityParallel} unless $N_1=N_2=\ldots = N_L$ (which is a trivial case for the parallel Gaussian channel). Essentially we cannot guarantee that $\sum_{k=1}^n X_{\ell,k}^2 = nP_\ell$ for all $\ell\in\mathcal{L}$ with probability $1$; we only know that the sum $\sum_{\ell\in\mathcal{L}}\sum_{k=1}^n X_{\ell,k}^2=nP$ with probability~$1$. Since we are able to conclude~\eqref{sumXk2=0whenL=1} from \eqref{powerEqualityParallel} for $L=1$ but unable to claim~\eqref{sumXk2=0whenL>1} from \eqref{powerEqualityParallel} for $L>1$, there is thus a discrepancy in the second- and third-order asymptotics between Theorem~\ref{thmMainResult} and Theorem~\ref{thmMainResultParallel} using the current proof technique.
%
%However, what we are able to show using the current technique is that the third-order term for the parallel Gaussian channel {\em without feedback} is upper bounded by $\frac{1}{2}\log n + O(1)$, improving on \cite[Th.~ 78]{Pol10} and matching the lower bound in \cite[Appendix~A]{TanTom13a}.
% Establishing the exact second- and third-order asymptotics for the parallel Gaussian channel {\em with feedback} is an avenue for future research.
%\section{Parallel AWGN Channel} \label{parallelAWGNChannelNoFeedback}
%We consider the parallel Gaussian  channel \cite[Chapter 9.4]{CoverBook} consisting of $L$~AWGN channels such that the following statement holds for each $k\in\{1, 2, \ldots, n\}$ and each $\ell\in \mathcal{L}$: In channel use~$k$, $\mathrm{s}$ transmits $X_{\ell,k}$, which is a function of $W$, on the $\ell^{\text{th}}$ channel and the corresponding channel output denoted by $Y_{\ell,k}$ is
%\[
%Y_{\ell,k} = X_{\ell,k} + Z_{\ell,k},
%\]
%where $\{Z_{\ell,k}\}_{\substack{k \in \{1, 2, \ldots, n\},\\ \ell \in \mathcal{L}}}$ are independent zero-mean Gaussian random variables such that the variance of $Z_{\ell,k}$ is $N_\ell$. The codewords $\{X_{\ell}^n\}_{\ell\in\mathcal{L}}$ transmitted by~$\mathrm{s}$ should satisfy the power constraint
% $\sum_{\ell=1}^L \sum_{k=1}^n X_{\ell,k}^2 \le n P$.
%In other words, $\Pr\{\sum_{\ell=1}^L \sum_{k=1}^n X_{\ell,k}^2 \le n P\}=1$. An $(n, M, P)$-code for the parallel Gaussian  channel is defined in a similar way to Definition~\ref{defCode}, except that $\{X_{\ell^n}\}_{\ell=1}^L$ is a function of $W$. To facilitate discussion, let $\boldsymbol{X}$ and $\boldsymbol{Y}$ denote the random variables $(X_1, X_2, \ldots, X_L)$ and $(Y_1, Y_2, \ldots, Y_L)$ respectively, and let $\mathbf{x}\stackrel{\text{def}}{=} (x_1, x_2, \ldots, x_L)$ and $\mathbf{y}\stackrel{\text{def}}{=} (y_1, y_2, \ldots, y_L)$ be the realizations of $\boldsymbol{X}$ and $\boldsymbol{Y}$ respectively. The parallel Gaussian  channel is characterized by $q_{\boldsymbol{Y}|\boldsymbol{X}}$ satisfying
%\[
%q_{\boldsymbol{Y}|\boldsymbol{X}}(\mathbf{y}|\mathbf{x})\stackrel{\text{def}}{=} \prod_{\ell=1}^L \phi_{0,N_\ell}(y_\ell-x_\ell)
%\]
%for all $\mathbf{x}\in \mathbb{R}^L$ and $\mathbf{y}\in\mathbb{R}^L$. We formally define the parallel Gaussian  channel and an $(n, M, P, \varepsilon)$-code in similar way to Definitions~\ref{defAWGNchannel} and \ref{defErrorProbability} respectively. The capacity of the parallel Gaussian  channel is well known and is achieved by the optimal power allocation among the~$L$ channels obtained from the water-filling algorithm \cite[Chapter 9.4]{CoverBook}, which yields $L+1$ real numbers denoted by $\Lambda$, $P_1$, $P_2$, $\ldots$, $P_L$ that satisfy
%\begin{equation}
%\sum_{\ell=1}^L P_\ell = P \label{sumPell=P}
%\end{equation}
%and
%\begin{equation}
%P_\ell = \max\{0,\Lambda - N_\ell\} \label{pEllValue}
%\end{equation}
%for each $\ell\in \mathcal{L}$. Recalling the definitions of $\mathrm{C}(P)$ and $\mathrm{V}(\mathbf{P}^*)$ in \eqref{defCP} and \eqref{defVP} respectively, we let
%\begin{equation}
%\mathrm{C}(\mathbf{P}^*) \stackrel{\text{def}}{=} \sum_{\ell=1}^L \mathrm{C}(P_\ell/N_\ell) \qquad\mbox{bits per channel use} \label{defCPparallelNFB}
%\end{equation}
%and
%\begin{equation}
%\mathrm{V}(\mathbf{P}^*) \stackrel{\text{def}}{=} \sum_{\ell=1}^L \mathrm{V}(P_\ell/N_\ell) \qquad\mbox{bits$^2$ per channel use}, \label{defVPparallelNFB}
%\end{equation}
%be the capacity and the dispersion of the parallel Gaussian  channel \cite[Th.~78]{Pol10} respectively. The following theorem states an upper bound on the first-, second- and third-order asymptotics for the parallel Gaussian  channel.
%\medskip
%\begin{Theorem} \label{thmMainResultParallelNoFeedback}
%Fix an $\varepsilon \in (0,1)$ and let
%\[
%M^*(n, \varepsilon, P) \stackrel{\text{def}}{=} \max\left\{M \left|\: \parbox[c]{2.8 in}{There exists an $(n, M, P, \varepsilon)$-code for the AWGN channel consisting of~$L$ parallel channels}\right.\right\}.
%\]
%Recall the convention of $\Lambda$, $P_1, P_2, \ldots, P_L$, $\mathrm{C}(\mathbf{P}^*)$ and $\mathrm{V}(\mathbf{P}^*)$ stated in \eqref{sumPell=P}, \eqref{pEllValue}, \eqref{defCPparallelNFB} and \eqref{defVPparallelNFB}.
%Then, there exists a constant $\kappa$ not depending on~$n$ such that for each $n\in \mathbb{N}$,
%\begin{equation}
%\log M^*(n, \varepsilon, P) \le \mathrm{C}(\mathbf{P}^*) + \sqrt{n\mathrm{V}(\mathbf{P}^*)}\Phi^{-1}(\varepsilon) +  \frac{1}{2}\log n + \kappa. \label{cutsetStatementParallelNFB}
%\end{equation}
%\end{Theorem}
%\begin{IEEEproof}
%Fix an $\varepsilon\in (0,1)$ and choose an arbitrary sequence of $(\bar n, M_{\mathrm{fb}}^*(\bar n, \varepsilon, P), P, \varepsilon)$-codes for the parallel Gaussian  channel with feedback. Letting $n=\bar n+1$ and following similar procedures for proving \eqref{eqnBHTSecondChain} from the beginning of the proof of Theorem~\ref{thmMainResult}, we obtain a sequence of $(n, M^*(n-1, \varepsilon, P), P, \varepsilon)$-codes with
% \begin{equation}
% \Pr\left\{\sum_{\ell=1}^L \sum_{k=1}^n X_{\ell,k}^2 = n P\right\}=1 \label{powerEqualityParallelNFB}
% \end{equation}
% such that the following inequality holds for each $n\in \{2, 3, \ldots\}$ and each $\xi_n >0$:
%\begin{align}
%&\log M^*(n-1, \varepsilon, P)\notag\\
%& \le \log\xi_n - \log\left(\Pr_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}} \left\{\sum_{k=1}^n \log\frac{q_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{Y}_{k}|\boldsymbol{X}_{k})}{r_{\boldsymbol{Y}_{k}}(\boldsymbol{Y}_{k})} < \log\xi_n \right\}-\varepsilon\right),\label{eqnBHTSecondChainParallelNFB}
%\end{align}
%where $p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}$ denotes the probability distribution induced by the $(n, M^*(n-1, \varepsilon, P), P, \varepsilon)$-feedback code, and $r_{\boldsymbol{Y}_{k}}$ is defined for each $k\in\{1, 2, \ldots, n\}$ as
%\begin{equation}
%r_{\boldsymbol{Y}_{k}}(\mathbf{y}_{k})=\prod_{\ell=1}^L \phi_{0, P_\ell + N_\ell}(y_{\ell,k}) \label{defsYkParallelNFB}
%\end{equation}
%for all $\mathbf{y}_{k}$. The channel law is
%\begin{equation}
%q_{\boldsymbol{Y}|\boldsymbol{X}}(\mathbf{y}_k|\mathbf{x}_k)=\prod_{\ell=1}^L \phi_{0,N_\ell}(y_{\ell,k}-x_{\ell,k}) \label{memorylessStatement++parallelNFB}
%\end{equation}
%for each $k\in\{1, 2, \ldots, n\}$. Combining \eqref{defsYkParallelNFB} and \eqref{memorylessStatement++parallelNFB}, we have
%\begin{align}
%& \log\frac{q_{\boldsymbol{Y}|\boldsymbol{X}}(\boldsymbol{Y}_k|\boldsymbol{X}_k)}{r_{\boldsymbol{Y}_k}(\boldsymbol{Y}_k)} \notag\\
%& = \sum_{\ell=1}^L \left( \frac{1}{2}\log\left(1+\frac{P_\ell}{N_\ell}\right)+ \frac{}{2(N_\ell+P_\ell)}\left(\frac{-P_\ell}{N_\ell}(Y_{\ell,k}-X_{\ell,k})^2 + X_{\ell,k}^2 + 2X_{\ell,k}(Y_{\ell,k}-X_{\ell,k})\right)\right)\notag\\
%&\stackrel{\text{(a)}}{=}\sum_{\ell=1}^L \left( \frac{1}{2}\log\left(1+\frac{P_\ell}{N_\ell}\right)+ \frac{}{2\Lambda}\left(\frac{-P_\ell}{N_\ell}(Y_{\ell,k}-X_{\ell,k})^2 + X_{\ell,k}^2 + 2X_{\ell,k}(Y_{\ell,k}-X_{\ell,k})\right)\right) \label{defLogLikelihoodParallelNFB}
%\end{align}
%for each $k\in\{1, 2, \ldots, n\}$, where (a) follows from \eqref{pEllValue}.
%Letting
%\begin{equation}
%U_{k} \stackrel{\text{def}}{=} \sum_{\ell=1}^L \left( \frac{1}{2}\log\left(1+\frac{P_\ell}{N_\ell}\right)+ \frac{}{2\Lambda}\left(\frac{-P_\ell}{N_\ell}(Y_{\ell,k}-X_{\ell,k})^2 + P_\ell + 2X_{\ell,k}(Y_{\ell,k}-X_{\ell,k})\right) \right)\label{defUkParallelNFB}
%\end{equation}
%for each $k\in\{1, 2, \ldots, n\}$,
%it follows from \eqref{defCPparallelNFB} and \eqref{powerEqualityParallelNFB} that
%\begin{equation*}
%\Pr\left\{\sum_{k=1}^n U_{k}= n \mathrm{C}(\mathbf{P}^*) +\sum_{k=1}^n \sum_{\ell=1}^L  \frac{}{2\Lambda}\left(\frac{-P_\ell}{N_\ell}(Y_{\ell,k}-X_{\ell,k})^2 + X_{\ell,k}^2 + 2X_{\ell,k}(Y_{\ell,k}-X_{\ell,k})\right)\right\}=1,
%\end{equation*}
%which implies from \eqref{eqnBHTSecondChainParallelNFB} and \eqref{defLogLikelihoodParallelNFB} that
%\begin{align}
%&\log M^*(n-1, \varepsilon, P)\notag\\
%& \le \log\xi_n - \log\left(\Pr_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}} \left\{\sum_{k=1}^n U_{k} < \log\xi_n - n \mathrm{C}(\mathbf{P}^*) \right\}-\varepsilon\right).\label{eqnBHT3rdChainParallelNFB}
%\end{align}
%Define the function
%$
%\lambda: \mathbb{R}^L\times \mathbb{R}^L\rightarrow \mathbb{R}
%$
%\begin{equation}
%\lambda(\mathbf{x}, \mathbf{y})= \sum_{\ell=1}^L -P(y_\ell-x_\ell)^2/N_\ell + 2 x_\ell(y_\ell-x_\ell). \label{deflambdaParallelNFB}
%\end{equation}
%We begin evaluating the distribution of $\sum_{k=1}^n U_{\ell,k}$ by examining the distribution of
%$\sum_{k=1}^n \lambda(\boldsymbol{X}_{k}, \boldsymbol{Y}_{k})$ (cf.\ \eqref{defUkParallelNFB}) as follows.
%Let
%\begin{equation}
%\E_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}}\left[e^{t\sum_{k=1}^n \lambda(\boldsymbol{X}_{k}, \boldsymbol{Y}_{k})}\right] \label{momentGenPartialUkParallelNFB}
%\end{equation}
% be the moment generating function of $\sum_{k=1}^n \lambda(\boldsymbol{X}_{k}, \boldsymbol{Y}_{k})$. In order to evaluate a closed form expression for \eqref{momentGenPartialUkParallelNFB}, we write
% \begin{align}
% &\E_{p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}}\left[e^{t\sum_{k=1}^n \lambda(\boldsymbol{X}_{k}, \boldsymbol{Y}_{k})}\right]  \notag\\
%&= \int_{w, \mathbf{x}^n, \mathbf{y}^n} p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}(w, \mathbf{x}^n, \mathbf{y}^n)e^{t\sum_{k=1}^n \lambda(\boldsymbol{X}_{k}, \boldsymbol{Y}_{k})}
%  \, \mathrm{d}\mathbf{y}^n \, \mathrm{d}\mathbf{x}^n\, \mathrm{d}w \notag\\
%  & \stackrel{\text{(a)}}{=} \int\limits_{w, \mathbf{x}^n, \mathbf{y}^n} p_{W, \boldsymbol{X}^n, \boldsymbol{Y}^n}(w, \mathbf{x}^n,\mathbf{y}^n)e^{t\left(\sum\limits_{k=1}^n \lambda(\boldsymbol{X}_{k}, \boldsymbol{Y}_{k})\right)+ \left(\sum\limits_{\ell=1}^L\frac{2t^2N_\ell}{1+2tP_\ell}\left(nP_\ell - \sum\limits_{k=1}^n x_{\ell,k}^2\right)\right)}
%  \, \mathrm{d}\mathbf{y}^n \, \mathrm{d}\mathbf{x}^n\, \mathrm{d}w \notag\\
%  &  \stackrel{\text{(b)}}{=}\frac{1}{|\mathcal{W}|}\sum\limits_{w\in \mathcal{W}}\int_{\mathbf{x}^n, \mathbf{y}^n} p_{\boldsymbol{X}^{n}, \boldsymbol{Y}^{n}|W}(\mathbf{x}^n,\mathbf{y}^n|w)e^{t\left(\sum\limits_{k=1}^n \lambda(\boldsymbol{X}_{k}, \boldsymbol{Y}_{k})\right)+ \left(\sum\limits_{\ell=1}^L\frac{2t^2}{1+2tP_\ell}\left(nP_\ell - \sum\limits_{k=1}^n x_{\ell,k}^2\right)\right)}
%  \, \mathrm{d}\mathbf{y}^n \, \mathrm{d}\mathbf{x}^n \label{eqnBHT4thChainParallelNFB}
%\end{align}
% where
% \begin{enumerate}
% \item[(a)] follows from \eqref{powerEqualityParallelNFB} that
%%\[
%%\Pr_{p_{W,X^n, Y^n}}\left\{\sum_{k=1}^n  \boldsymbol{X}_k^2 = nP \right\}=1.
%%\]
%\[
%\frac{2t^2}{1+2tP} \left(nP - \sum\limits_{k=1}^{n}x_k^2\right)=0
%\]
%for all $\mathbf{x}^n$ on the support of $p_{\boldsymbol{X}^n}$.
%\item[(b)] follows from the fact that $W$ is uniform on $\mathcal{W}$.
%\end{enumerate}
%\end{IEEEproof}
%\section{Conclusion} \label{conclusion}


% conference papers do not normally have an appendix

% use section* for acknowledgement

% optional entry into table of contents (if used)
%\addcontentsline{toc}{section}{Acknowledgment}
%The work of Gerhard Kramer was supported by the German Ministry of Education and Research and the Alexander von Humboldt Foundation in the framework of the Alexander von Humboldt Professorship.

% An example of a floating figure using the graphicx package.
% Note that \label must occur AFTER (or within) \caption.
% For figures, \caption should occur after the \includegraphics.
% Note that IEEEtran v1.7 and later has special internal code that
% is designed to preserve the operation of \label within \caption
% even when the captionsoff option is in effect. However, because
% of issues like this, it may be the safest practice to put all your
% \label just after \caption rather than within \caption{}.
%
% Reminder: the "draftcls" or "draftclsnofoot", not "draft", class
% option should be used if it is desired that the figures are to be
% displayed while in draft mode.
%
%\begin{figure}[!t]
%\centering
%\includegraphics[width=2.5in]{myfigure}
% where an .eps filename suffix will be assumed under latex,
% and a .pdf suffix will be assumed for pdflatex; or what has been declared
% via \DeclareGraphicsExtensions.
%\caption{Simulation Results}
%\label{fig_sim}
%\end{figure}

% Note that IEEE typically puts floats only at the top, even when this
% results in a large percentage of a column being occupied by floats.


% An example of a double column floating figure using two subfigures.
% (The subfig.sty package must be loaded for this to work.)
% The subfigure \label commands are set within each subfloat command, the
% \label for the overall figure must come after \caption.
% \hfil must be used as a separator to get equal spacing.
% The subfigure.sty package works much the same way, except \subfigure is
% used instead of \subfloat.
%
%\begin{figure*}[!t]
%\centerline{\subfloat[Case I]\includegraphics[width=2.5in]{subfigcase1}%
%\label{fig_first_case}}
%\hfil
%\subfloat[Case II]{\includegraphics[width=2.5in]{subfigcase2}%
%\label{fig_second_case}}}
%\caption{Simulation results}
%\label{fig_sim}
%\end{figure*}
%
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% captions (using the optional argument to \subfloat), but instead will
% reference/describe all of them (a), (b), etc., within the main caption.


% An example of a floating table. Note that, for IEEE style tables, the
% \caption command should come BEFORE the table. Table text will default to
% \footnotesize as IEEE normally uses this smaller font for tables.
% The \label must come after \caption as always.
%
%\begin{table}[!t]
%% increase table row spacing, adjust to taste
%\renewcommand{\arraystretch}{1.3}
% if using array.sty, it might be a good idea to tweak the value of
% \extrarowheight as needed to properly center the text within the cells
%\caption{An Example of a Table}
%\label{table_example}
%\centering
%% Some packages, such as MDW tools, offer better commands for making tables
%% than the plain LaTeX2e tabular which is used here.
%\begin{tabular}{|c\|c|}
%\hline
%One & Two\\
%\hline
%Three & Four\\
%\hline
%\end{tabular}
%\end{table}


% Note that IEEE does not put floats in the very first column - or typically
% anywhere on the first page for that matter. Also, in-text middle ("here")
% positioning is not used. Most IEEE journals use top floats exclusively.
% Note that, LaTeX2e, unlike IEEE journals, places footnotes above bottom
% floats. This can be corrected via the \fnbelowfloat command of the
% stfloats package.




% if have a single appendix:
%\appendix
%\begin{IEEEproof}[Proof of Lemma~\ref{lemmaDPI}]
%In order to prove Statement 1, we fix a function $g$, and let $\mathcal{Y} = \{g(x): x\in \mathcal{X}\}$. Using Definition~\ref{defBHTDivergence}, we have
%\begin{align*}
%& \beta_{\delta}(p_{g(X)}\|q_{g(X)}) \notag\\
%&= \inf\limits_{\substack{r_{Z|Y} \in \mathcal{A}(\{0,1\}|\mathcal{Y}): \\ \int_{y\in \mathcal{Y}}r_{Z|Y}(1|y)p_{g(X)}(y)\, \mathrm{d}y\ge \delta}} \int_{y\in \mathcal{Y}}r_{Z|Y}(1|y)q_{g(X)}(y)\, \mathrm{d}y \notag\\
%&= \inf\limits_{\substack{r_{Z|Y} \in \mathcal{A}(\{0,1\}|\mathcal{Y}): \\ \int_{y\in \mathcal{Y}}r_{Z|Y}(1|y)\int_{x\in\mathcal{X}: g(x)=y}p_{X}(x)\, \mathrm{d}x\, \mathrm{d}y\ge \delta}} \int_{y\in \mathcal{Y}}r_{Z|Y}(1|y)\int_{x\in\mathcal{X}: g(x)=y}q_{X}(x)\, \mathrm{d}x\, \mathrm{d}y \notag\\
%&= \inf\limits_{\substack{r_{Z|Y} \in \mathcal{A}(\{0,1\}|\mathcal{Y}): \\\int_{x\in\mathcal{X}} \int_{y\in \mathcal{Y}:y=g(x)}r_{Z|Y}(1|y)p_{X}(x)\, \mathrm{d}y\, \mathrm{d}x\ge \delta}}\int_{x\in\mathcal{X}} \int_{y\in \mathcal{Y}:y=g(x)}r_{Z|Y}(1|y)q_{X}(x)\, \mathrm{d}y\, \mathrm{d}x\notag\\
%&= \inf\limits_{\substack{r_{Z|Y} \in \mathcal{A}(\{0,1\}|\mathcal{Y}): \\\int_{x\in\mathcal{X}} \int_{y\in \mathcal{Y}:y=g(x)}r_{Z|Y}(1|y)\, \mathrm{d}yp_{X}(x)\, \mathrm{d}x\ge \delta}}\int_{x\in\mathcal{X}} \int_{y\in \mathcal{Y}:y=g(x)}r_{Z|Y}(1|y)\, \mathrm{d}yq_{X}(x)\, \mathrm{d}x\notag\\
%& \ge \inf\limits_{\substack{r_{Z|X} \in \mathcal{A}(\{0,1\}|\mathcal{X}): \\ \int_{x\in \mathcal{X}}r_{Z|X}(1|x)p_X(x)\, \mathrm{d}x\ge \delta}} \int_{x\in \mathcal{X}}r_{Z|X}(1|x)q_X(x)\, \mathrm{d}x \notag\\
%& = \beta_{\delta}(p_X\|q_X).
%\end{align*}
%The proof of Statement 2 is contained in \cite[Section 2.3]{Pol10} and therefore omitted.
%%, we let $\xi_n>0$ be an arbitrary real number. Then,
%%\begin{align*}
%%&\beta_{\delta}(p_X\|q_X) \notag\\
%%& = \inf\limits_{\substack{r_{Z|X} \in \mathcal{A}(\{0,1\}|\mathcal{X}): \\ \int_{x\in \mathcal{X}}r_{Z|X}(1|x)p_X(x)\, \mathrm{d}x\ge \delta}} \int_{x\in \mathcal{X}}r_{Z|X}(1|x)q_X(x)\, \mathrm{d}x \notag\\
%%& \ge \inf\limits_{\substack{r_{Z|X} \in \mathcal{A}(\{0,1\}|\mathcal{X}): \\ \int_{x\in \mathcal{X}}r_{Z|X}(1|x)p_X(x)\, \mathrm{d}x\ge \delta}} \int_{x\in \mathcal{X}}r_{Z|X}(1|x)q_X(x) \boldsymbol{1}\left(\left\{\frac{p_X(x)}{q_X(x)}< \xi \right\}\right)\, \mathrm{d}x \notag\\
%%& \ge \inf\limits_{\substack{r_{Z|X} \in \mathcal{A}(\{0,1\}|\mathcal{X}): \\ \int_{x\in \mathcal{X}}r_{Z|X}(1|x)p_X(x)\, \mathrm{d}x\ge \delta}} \frac{1}{\xi} \int_{x\in \mathcal{X}}r_{Z|X}(1|x)p_X(x) \boldsymbol{1}\left(\left\{\frac{p_X(x)}{q_X(x)}< \xi \right\}\right)\, \mathrm{d}x  \notag\\
%%& = \inf\limits_{\substack{r_{Z|X} \in \mathcal{A}(\{0,1\}|\mathcal{X}): \\ \int_{x\in \mathcal{X}}r_{Z|X}(1|x)p_X(x)\, \mathrm{d}x\ge \delta}} \frac{1}{\xi} \int_{x\in \mathcal{X}}r_{Z|X}(1|x)p_X(x) \left(1- \boldsymbol{1}\left(\left\{\frac{p_X(x)}{q_X(x)}\ge \xi \right\}\right)\right)\, \mathrm{d}x \notag\\
%%&\ge \frac{1}{\xi}\left(\delta- \int_{x\in \mathcal{X}}p_X(x)\boldsymbol{1}\left(\left\{\frac{p_X(x)}{q_X(x)}\ge \xi \right\}\right)\, \mathrm{d}x \right).
%%\end{align*}
%\end{IEEEproof}
%\medskip
%\begin{IEEEproof}[Proof of Proposition~\ref{propositionBHTLowerBound}]
%The proof technique is essentially the same as that in \cite[Th.~27]{PPV10}. We provide the proof here for the sake of completeness. Let $s_{U,V}\stackrel{\text{def}}{=} p_Uq_V$. Consider the following chain of inequalities:
%\begin{align*}
% \beta_{1-\alpha}(p_{U,V}\|s_{U,V})
% & \stackrel{\text{(a)}}{=} \inf\limits_{\substack{r_{Z|U,V} \in \mathcal{A}(\{0,1\}|\mathcal{U}\times \mathcal{V}): \\ \sum_{(u,v)\in \mathcal{U}\times \mathcal{V}}r_{Z|U,V}(1|u,v)p_{U,V}(u,v)\ge 1-\alpha}} \sum_{(u,v)\in \mathcal{U}\times \mathcal{V}}r_{Z|U,V}(1|u,v)q_{U,V}(u,v) \notag \\
% & \stackrel{\text{(b)}}{\le} \sum_{(u,v)\in \mathcal{U}\times \mathcal{V}}\boldsymbol{1}(\{u=v\})q_{U,V}(u,v) \notag \\
%&\stackrel{\text{(c)}}{=} 1/|\mathcal{W}|,
%\end{align*}
%where
%\begin{enumerate}
%\item[(a)] follows from the DPI in Lemma~\ref{lemmaDPI};
%\item[(b)] follows from \eqref{defAlpha} that
%\begin{align*}
%\sum_{u,v}\boldsymbol{1}(\{u=v\}) p_{U,V}(u,v) = 1-\alpha;
%\end{align*}
%\item[(c)] follows from the facts that $s_{U,V}= p_Uq_V$ and $p_U(u) = 1/|\mathcal{W}|$ for all $u\in\mathcal{W}$.
%\end{enumerate}
%This completes the proof.
%\end{IEEEproof}
% or
%\appendices
%\section{Proof of Lemma~\ref{lemmaDPI}}\label{appendixA}


 % for no appendix heading
% do not use \section anymore after \appendix, only \section*
% is possibly needed



% use appendices with more than one appendix
% then use \section to start each appendix
% you must declare a \section before using any
% \subsection or using \label (\appendices by itself
% starts a section numbered zero.)
%
\appendices
\section{Proof of Lemma~\ref{lemmaTransformedCode}}\label{appendixA}
\begin{IEEEproof}
Let $f_{\ell,k}$ and $\tilde f_{\ell,k}$ be the encoding functions of the $(n, M, P)$-feedback code and the $(\gamma, \mathbf{s})$-modified code respectively for each $\ell\in\mathcal{L}$ and each $k\in\{1, 2, \ldots, n\}$. For any $w\in\mathcal{W}$ and any $\mathbf{y}^n\in \mathbb{R}^{L\times n}$ such that
\begin{align}
\phi\left( \left[\begin{matrix}
f_{1,1}(w) \\\vdots\\ f_{L,1}(w)
\end{matrix} \right]
,\ldots, \left[\begin{matrix}
f_{1,n}(w, \mathbf{y}^{n-1}) \\\vdots\\ f_{L,n}(w, \mathbf{y}^{n-1})
\end{matrix} \right]\right)\in \Gamma^{(\gamma)}(\mathbf{s}) \label{lemmaTransformedCodeEq1}
\end{align}
and
\begin{align}
\sum_{\ell=1}^L\sum_{k=1}^n f_{\ell,k}(w, \mathbf{y}^{k-1})^2=nP, \label{lemmaTransformedCodeEq2}
\end{align}
it follows from~\eqref{defTildeFeq1}, \eqref{defTildeFeq2} and~\eqref{defTildeFeq3} in Definition~\ref{definitionTransformedCode}
 %and the assumption $\sum_{\ell=1}^L s_\ell = P$
 that
\begin{align}
\left( \left[\begin{matrix}
f_{1,1}(w) \\\vdots\\ f_{L,1}(w)
\end{matrix} \right],\ldots,
\left[\begin{matrix}
f_{1,n}(w, \mathbf{y}^{n-1}) \\\vdots\\ f_{L,n}(w, \mathbf{y}^{n-1})
\end{matrix} \right]\right)=
\left( \left[\begin{matrix}
\tilde f_{1,1}(w) \\\vdots\\ \tilde f_{L,1}(w)
\end{matrix} \right]
,\ldots, \left[\begin{matrix}
\tilde f_{1,n}(w, \mathbf{y}^{n-1}) \\\vdots\\ \tilde f_{L,n}(w, \mathbf{y}^{n-1})
\end{matrix} \right]\right). \label{lemmaTransformedCodeEq3}
\end{align}
Since~\eqref{lemmaTransformedCodeEq3} holds for any $w\in\mathcal{W}$ and $\mathbf{y}^n\in \mathbb{R}^{L\times n}$ that satisfy~\eqref{lemmaTransformedCodeEq1} and~\eqref{lemmaTransformedCodeEq2}, it follows that~\eqref{lemmaTransformedCodeSt} holds for all Borel measurable~$\mathcal{A}\subseteq\mathbb{R}^{L\times n}\times \mathbb{R}^{L\times n}$.
\end{IEEEproof}


\section{Proof of Lemma~\ref{lemma1InProof}}\label{appendixC}
%Fix any $\gamma\ge 0$ and any $t\in\mathbb{R}$. %It suffices to show that
% \begin{align}
% \E_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{*}}\left[e^{t\sum\limits_{k=1}^n U_k^{(\mathbf{P}^*)}}\cdot e^{t^2\sum\limits_{\ell=1}^L\frac{N_\ell \left(nP_\ell  -\sum\limits_{k=1}^n X_{\ell,k}^2\right) }{2(P_\ell+N_\ell)\left((1+t)P_\ell+N_\ell\right)}}\right] = \E_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{*}}\left[e^{t\sum\limits_{k=1}^n V_k^{(\mathbf{P}^*)}}\right], \label{appendixCeq1}
% \end{align}
% which will then imply by using~\eqref{levyThmEq1*} that~\eqref{levyThmEq2} holds. To this end,
The proof for the AWGN channel case (i.e., $L=1$) is contained in~\cite[Sec.~E]{FongTan15}. For a general~$L\in \mathbb{N}$, consider the following chain of equalities for each $m=n, n-1, \ldots, 1$:
 \begin{align}
& \E\left[e^{\lambda \sum\limits_{k=1}^m U_k^{(\mathbf{P}^*)}}\cdot e^{\lambda ^2\sum\limits_{\ell=1}^L\frac{N_\ell \left(nP_\ell  -\sum\limits_{k=1}^m X_{\ell,k}^2\right) }{2(P_\ell+N_\ell)\left((1+\lambda )P_\ell+N_\ell\right)}}\right] \notag\\
& =  \E\left[\E\left[\left.e^{\lambda\sum\limits_{k=1}^{m} U_k^{(\mathbf{P}^*)}}\cdot e^{\lambda^2\sum\limits_{\ell=1}^L\frac{N_\ell \left(nP_\ell  -\sum\limits_{k=1}^{m} X_{\ell,k}^2\right) }{2(P_\ell+N_\ell)\left((1+\lambda  )P_\ell+N_\ell\right)}}\right|\boldsymbol{X}^{m}, \boldsymbol{Z}^{m-1}\right]\right] \\
& = \E\left[ e^{\lambda\sum\limits_{k=1}^{m-1} U_k^{(\mathbf{P}^*)}}\cdot e^{\lambda^2\sum\limits_{\ell=1}^L\frac{N_\ell \left(nP_\ell  -\sum\limits_{k=1}^{m-1} X_{\ell,k}^2\right) }{2(P_\ell+N_\ell)\left((1+\lambda  )P_\ell+N_\ell\right)}} \cdot \int_{\mathbb{R}^{L\times n}}p_{\boldsymbol{Z}_m}^*(\mathbf{z}_m) \cdot e^{\lambda U_m^{(\mathbf{P}^*)}}\cdot e^{\lambda^2\sum\limits_{\ell=1}^L \frac{- X_{\ell,m}^2 }{2(P_\ell+N_\ell)\left((1+\lambda  )P_\ell+N_\ell\right)}} \mathrm{d}\mathbf{z}_m \right] \label{appendixCeq2b-}\\
&= e^{\lambda\sum\limits_{\ell=1}^L\frac{P_\ell}{2(P_\ell+N_\ell)}}\sqrt{\prod_{\ell=1}^L\frac{P_\ell+N_\ell}{(1+\lambda  )P_\ell+N_\ell}} \cdot \E\left[e^{\lambda\sum\limits_{k=1}^{m-1} U_k^{(\mathbf{P}^*)}}\cdot e^{\lambda^2\sum\limits_{\ell=1}^L\frac{N_\ell \left(nP_\ell  -\sum\limits_{k=1}^{m-1} X_{\ell,k}^2\right) }{2(P_\ell+N_\ell)\left((1+\lambda  )P_\ell+N_\ell\right)}}\right] \label{appendixCeq2b}
 \end{align}
 where 
 \begin{itemize}
 \item  \eqref{appendixCeq2b-} is due to the fact that $\boldsymbol{Z}_m$ and $(\boldsymbol{X}^{m},\boldsymbol{Z}^{m-1})$ are independent.
     \item \eqref{appendixCeq2b} is due to the definition of~$U_k^{(\mathbf{P}^*)}$ in~\eqref{defUkd}.
 \end{itemize}
 Applying \eqref{appendixCeq2b} recursively from $m=n$ to $m=1$, we have
  \begin{align}
  & \E\left[e^{\lambda\sum\limits_{k=1}^n U_k^{(\mathbf{P}^*)}}\cdot e^{\lambda^2\sum\limits_{\ell=1}^L\frac{N_\ell \left(nP_\ell  -\sum\limits_{k=1}^n X_{\ell,k}^2\right) }{2(P_\ell+N_\ell)\left((1+\lambda  )P_\ell+N_\ell\right)}}\right] \notag\\
& =\left(\prod_{\ell=1}^L\frac{P_\ell+N_\ell}{(1+\lambda  )P_\ell+N_\ell}\right)^{n/2}e^{n\sum\limits_{\ell=1}^L\left(\frac{\lambda P_\ell}{2(P_\ell+N_\ell)}+\frac{\lambda^2N_\ell P_\ell}{2(P_\ell+N_\ell)\left((1+\lambda  )P_\ell+N_\ell\right)}\right)}. \label{appendixCeq3}
  \end{align}
  On the other hand, straightforward calculations based on  the definition of~$V_k^{(\mathbf{P}^*)}$ in~\eqref{defVkd} and the fact that $\{\boldsymbol{Z}_k\}_{k=1}^n$ are independent implies that
 \begin{align}
 \E\left[e^{\lambda\sum\limits_{k=1}^n V_k^{(\mathbf{P}^*)}}\right]
& =\left(\prod_{\ell=1}^L\frac{P_\ell+N_\ell}{(1+\lambda  )P_\ell+N_\ell}\right)^{n/2}e^{n\sum\limits_{\ell=1}^L\left(\frac{\lambda P_\ell}{2(P_\ell+N_\ell)}+\frac{\lambda^2N_\ell P_\ell}{2(P_\ell+N_\ell)\left((1+\lambda  )P_\ell+N_\ell\right)}\right)}. \label{appendixCeq4}
 \end{align}
 Combining~\eqref{appendixCeq3} and~\eqref{appendixCeq4}, we obtain~\eqref{appendixCeq1}.
 %\begin{align}
% \E_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{*}}\left[e^{t\sum\limits_{k=1}^n U_k^{(\mathbf{P}^*)}}\cdot e^{t^2\sum\limits_{\ell=1}^L\frac{N_\ell \left(nP_\ell  -\sum\limits_{k=1}^n X_{\ell,k}^2\right) }{2(P_\ell+N_\ell)\left((1+t)P_\ell+N_\ell\right)}}\right]
% &=\left(\prod_{\ell=1}^L\frac{P_\ell+N_\ell}{(1+t)P_\ell+N_\ell}\right)^{n/2}e^{n\sum\limits_{\ell=1}^L\left(\frac{tP_\ell}{2(P_\ell+N_\ell)}+\frac{t^2N_\ell P_\ell}{2(P_\ell+N_\ell)\left((1+t)P_\ell+N_\ell\right)}\right)}\\
%  &= \E_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{*}}\left[e^{t\sum\limits_{k=1}^n V_k^{(\mathbf{P}^*)}}\right]. \label{appendixCeq5}
% \end{align}

 \section{Derivations of~\eqref{eqnTaylor1st} and~\eqref{eqnTaylor2nd}} \label{appendixB}
Straightforward calculations based on~\eqref{defFunctionF} reveal that for all $\mathbf{s}\in \mathbb{R}_+^L$, we obtain that
\begin{align}
\triangledown f(\mathbf{s}) = \frac{1}{2}\left[\begin{matrix}\frac{1}{N_1 + s_1}-\frac{1}{2\Lambda}+\mu_1 \\ \vdots\\  \frac{1}{N_L + s_L}-\frac{1}{2\Lambda}+\mu_L \end{matrix}\right] \label{appendixBeq1}
\end{align}
and $\triangledown^2 f(\mathbf{s})$ is a diagonal matrix that satisfies
\begin{align}
\triangledown^2 f(\mathbf{s}) = \frac{-1}{2}\left[\begin{matrix}\frac{1}{(N_1 + s_1)^2} & 0 & \ldots & 0\\ 0& \frac{1}{(N_2 + s_2)^2} &\ldots & 0\\\vdots & \vdots & \ddots &  \vdots\\0&0&\ldots& \frac{1}{(N_L + s_L)^2}\end{matrix}\right].  \label{appendixBeq2}
\end{align}
Combining~\eqref{appendixBeq1}, \eqref{sumPell=P}, \eqref{pEllValue} and~\eqref{defMu}, we have~$\triangledown f(\mathbf{P}^*) = 0$. In addition, for any $\mathbf{s}$ such that $\sum_{\ell=1}^L s_\ell \le P$, it follows from~\eqref{appendixBeq2} that $N_\ell + s_\ell \le N_{\max} +P$ for all $\ell\in\mathcal{L}$, which then implies that~\eqref{eqnTaylor2nd} holds for all $\mathbf{s}\in \mathcal{S}^{(n)}$.

\section{Derivation of~\eqref{eqnBHT9thChaina}}\label{appendixD}
Let $t=\frac{1}{\sqrt{n}}$. Fix any $\mathbf{s}\in \mathcal{S}^{(n)}\setminus\Pi^{(n)}$. Due to~\eqref{defDistPs}, it suffices to show that
 \begin{align}
&\E_{p_{\boldsymbol{X}^n, \boldsymbol{Y}^n}^{(\mathbf{s})}}\left[e^{t\sum\limits_{k=1}^n U_k^{(\mathbf{s})}}\cdot e^{t^2\sum\limits_{\ell=1}^L\frac{N_\ell \left(ns_\ell  -\sum\limits_{k=1}^n X_{\ell,k}^2\right) }{2(s_\ell+N_\ell)\left((1+t)s_\ell+N_\ell\right)}}\right] \notag\\
 &= \left(\prod_{\ell=1}^L\frac{s_\ell+N_\ell}{(1+t)s_\ell+N_\ell}\right)^{n/2}e^{n\sum\limits_{\ell=1}^L\left(\frac{ts_\ell}{2(s_\ell+N_\ell)}+\frac{t^2N_\ell s_\ell}{2(s_\ell+N_\ell)\left((1+t)s_\ell+N_\ell\right)}\right)}. \label{appendixDeq1}
 \end{align}
Replacing $\mathbf{P}^*$ with $\mathbf{s}$ in the steps leading to~\eqref{appendixCeq2b} and~\eqref{appendixCeq3}, we obtain~\eqref{appendixDeq1}.
%for each $m=n, n-1, \ldots, 1$
% \begin{align}
%& \E\left[e^{t\sum\limits_{k=1}^m U_k^{(\mathbf{P}^*)}}\cdot e^{t^2\sum\limits_{\ell=1}^L\frac{N_\ell \left(ns_\ell  -\sum\limits_{k=1}^m X_{\ell,k}^2\right) }{2(s_\ell+N_\ell)\left((1+t)s_\ell+N_\ell\right)}}\right] \notag\\
%& =  \E\left[\E\left[\left.e^{t\sum\limits_{k=1}^{m} U_k^{(\mathbf{P}^*)}}\cdot e^{t^2\sum\limits_{\ell=1}^L\frac{N_\ell \left(ns_\ell  -\sum\limits_{k=1}^{m} X_{\ell,k}^2\right) }{2(s_\ell+N_\ell)\left((1+t)s_\ell+N_\ell\right)}}\right|\boldsymbol{X}^{m}, \boldsymbol{Z}^{m-1}\right]\right] \\
%&= e^{t\sum\limits_{\ell=1}^L\frac{s_\ell}{2(s_\ell+N_\ell)}}\sqrt{\prod_{\ell=1}^L\frac{s_\ell+N_\ell}{(1+t)s_\ell+N_\ell}} \cdot\E\left[\E\left[\left.e^{t\sum\limits_{k=1}^{m-1} U_k^{(\mathbf{P}^*)}}\cdot e^{t^2\sum\limits_{\ell=1}^L\frac{N_\ell \left(ns_\ell  -\sum\limits_{k=1}^{m-1} X_{\ell,k}^2\right) }{2(s_\ell+N_\ell)\left((1+t)s_\ell+N_\ell\right)}}\right|\boldsymbol{X}^{m-1}, \boldsymbol{Z}^{m-1}\right] \right] \label{appendixDeq2b}
% \end{align}
% which will then imply by using~\eqref{levyThmEq1*} that~\eqref{levyThmEq2} holds.
%\section*{Acknowledgment}

%\section{Proof of the First Zonklar Equation}
%Appendix one text goes here.

% you can choose not to have a title for an appendix
% if you want by leaving the argument blank
%\section{}
%Appendix two text goes here.




% Can use something like this to put references on a page
% by themselves when using endfloat and the captionsoff option.
\ifCLASSOPTIONcaptionsoff
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\fi



% trigger a \newpage just before the given reference
% number - used to balance the columns on the last page
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% the document is modified later
%\IEEEtriggeratref{8}
% The "triggered" command can be changed if desired:
%\IEEEtriggercmd{\enlargethispage{-5in}}

% references section

% can use a bibliography generated by BibTeX as a .bbl file
% BibTeX documentation can be easily obtained at:
% http://www.ctan.org/tex-archive/biblio/bibtex/contrib/doc/
% The IEEEtran BibTeX style support page is at:
% http://www.michaelshell.org/tex/ieeetran/bibtex/
%\bibliographystyle{IEEEtran}
% argument is your BibTeX string definitions and bibliography database(s)
%\bibliography{IEEEabrv,../bib/paper}
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% <OR> manually copy in the resultant .bbl file
% set second argument of \begin to the number of references
% (used to reserve space for the reference number labels box)
%\begin{thebibliography}{1}
%
%\bibitem{IEEEhowto:kopka}
%H.~Kopka and P.~W. Daly, \emph{A Guide to \LaTeX}, 3rd~ed.\hskip 1em plus
% 0.5em minus 0.4em\relax Harlow, England: Addison-Wesley, 1999.
%
%\end{thebibliography}

\section*{Acknowledgments}
The authors would like to thank the Associate Editor Prof.\ Shun Watanabe and the two anonymous reviewers for the useful comments that improve the presentation of this paper.% in an earlier draft of this manuscript.
%
%The authors gratefully acknowledge financial support from  the National University of Singapore (NUS) under startup grant R-263-000-A98-750/133 and  NUS Young Investigator Award   R-263-000-B37-133.
%
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\bibitem{CoverBook}
{T. M. Cover and J. A. Thomas}, \emph{{Elements of Information Theory}},
  2nd~ed.\hskip 1em plus 0.5em minus 0.4em\relax Hoboken, NJ: John Wiley and
  Sons, 2006.

\bibitem{elgamal}
A.~{El~Gamal} and Y.-H. Kim, \emph{Network Information Theory}.\hskip 1em plus
  0.5em minus 0.4em\relax Cambridge, U.K.: Cambridge University Press, 2012.

\bibitem{davidTseBook}
D.~Tse and P.~Viswanath, \emph{Fundamentals of Wireless Communication}.\hskip
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  2005.

\bibitem{Sha49}
C.~E. Shannon, ``Communication in the presence of noise,'' \emph{Proceedings of
  IRE}, vol.~37, no.~1, pp. 10--21, 1949.

\bibitem{Pol10}
Y.~Polyanskiy, ``Channel coding: {Non}-asymptotic fundamental limits,'' Ph.D.
  dissertation, Princeton University, 2010.

\bibitem{TanTom13a}
V.~Y.~F. Tan and M.~Tomamichel, ``The third-order term in the normal
  approximation for the {AWGN} channel,'' \emph{{IEEE} Trans. Inf. Theory},
  vol.~61, no.~5, pp. 2430--2438, 2015.

\bibitem{Sha56}
C.~E. Shannon, ``The zero error capacity of a noisy channel,'' \emph{IRE
  Transactions on Information Theory}, vol.~2, no.~3, pp. 8--19, 1956.

\bibitem{AW14}
Y.~Altu\u{g} and A.~B. Wagner, ``Feedback can improve the second-order coding
  performance in discrete memoryless channels,'' in \emph{Proc. IEEE Intl.
  Symp. Inf. Theory}, Honolulu, HI, Jul 2014, pp. 2361--2365.

\bibitem{PPV11b}
Y.~Polyanskiy, H.~V. Poor, and S.~Verd\'{u}, ``Feedback in the non-asymptotic
  regime,'' \emph{{IEEE} Trans. Inf. Theory}, vol.~57, no.~8, pp. 4903--4925,
  2011.

\bibitem{Machkouri}
M.~E. Machkouri and L.~Ouchti, ``Exact convergence rates in the central limit
  theorem for a class of martingales,'' \emph{Bernoulli}, vol.~13, no.~4, pp.
  981--999, Nov. 2007.

\bibitem{feller}
W.~Feller, \emph{An Introduction to Probability Theory and Its Applications},
  2nd~ed.\hskip 1em plus 0.5em minus 0.4em\relax Hoboken, NJ: John Wiley and
  Sons, 1971, vol.~2.

\bibitem{FongTan15}
{S.~L.~Fong and V.~Y.~F.~Tan}, ``{Asymptotic expansions for the {AWGN} channel
  with feedback under a peak power constraint},'' in \emph{Proc. IEEE Intl.
  Symp. Inf. Theory}, Hong Kong, Jun. 2015, pp. 311--315.

\bibitem{TFT17}
L.~V. Truong, S.~L. Fong, and V.~Y.~F. Tan, ``On {Gaussian} channels with
  feedback under expected power constraints and with non-vanishing error
  probabilities,'' \emph{{IEEE} Trans. Inf. Theory}, vol.~63, no.~3, pp.
  1746--1765, 2017.

\bibitem{Wang09}
L.~Wang, R.~Colbeck, and R.~Renner, ``Simple channel coding bounds,'' in
  \emph{Proc. IEEE Intl. Symp. Inf. Theory}, Seoul, South Korea, June/July
  2009, pp. 1804 -- 1808.

\bibitem{PPV10}
Y.~Polyanskiy, H.~V. Poor, and S.~Verd\'{u}, ``Channel coding rate in the
  finite blocklength regime,'' \emph{{IEEE} Trans. Inf. Theory}, vol.~56,
  no.~5, pp. 2307--2359, 2010.

\bibitem{curtiss}
J.~H. Curtiss, ``A note on the theory of moment generating functions,''
  \emph{Ann. Math. Statist.}, vol.~13, no.~4, pp. 430--433, 1942.

\bibitem{Williams1991}
D.~Williams, \emph{Probability with Martingales}.\hskip 1em plus 0.5em minus
  0.4em\relax Cambridge, U.K.: Cambridge University Press, 1991.

\bibitem{Hayashi09}
M.~Hayashi, ``Information spectrum approach to second-order coding rate in
  channel coding,'' \emph{{IEEE} Trans. Inf. Theory}, vol.~55, no.~11, pp.
  4947--4966, 2009.

\bibitem{TomTan12}
M.~Tomamichel and V.~Y.~F. Tan, ``A tight upper bound for the third-order
  asymptotics of most discrete memoryless channels,'' \emph{{IEEE} Trans. Inf.
  Theory}, vol.~59, no.~11, pp. 7041--7051, 2013.

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I.~A. Ibragimov and Y.~V. Linnik, \emph{Independent and stationary sequences of
  random variables}, J.~F.~C. Kingman, Ed.\hskip 1em plus 0.5em minus
  0.4em\relax Groningen, Netherlands: Wolters-Noordhoff Publishing, 1971.

\bibitem{Csi97}
I.~Csisz\'{a}r and J.~{K\"{o}rner}, \emph{Information Theory: Coding Theorems
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V.~Strassen, ``{Asymptotische Absch\"{a}tzungen in Shannons
  Informationstheorie},'' in \emph{Trans. Third Prague Conf. Inf. Theory},
  Prague, 1962, pp. 689--723,
  http://www.math.cornell.edu/$\sim$pmlut/strassen.pdf.

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%
%\bibitem{elgamal}
%A.~{El~Gamal} and Y.-H. Kim, \emph{Network Information Theory}.\hskip 1em plus
%  0.5em minus 0.4em\relax Cambridge, U.K.: Cambridge University Press, 2012.
%
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%Y.~Polyanskiy, ``Channel coding: {Non}-asymptotic fundamental limits,'' Ph.D.
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%
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%V.~Y.~F. Tan and M.~Tomamichel, ``The third-order term in the normal
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%  2013.
%
%\bibitem{PPV11b}
%Y.~Polyanskiy, H.~V. Poor, and S.~Verd\'{u}, ``Feedback in the non-asymptotic
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%
%\bibitem{feller}
%W.~Feller, \emph{An Introduction to Probability Theory and Its Applications},
%  2nd~ed.\hskip 1em plus 0.5em minus 0.4em\relax John Wiley and Sons, 1971.
%
%\bibitem{Wang09}
%L.~Wang, R.~Colbeck, and R.~Renner, ``Simple channel coding bounds,'' in
%  \emph{Proc. IEEE Intl. Symp. Inf. Theory}, Seoul, S. Korea, Jul. 2009, pp.
%  1804--1808.
%
%\bibitem{PPV10}
%Y.~Polyanskiy, H.~V. Poor, and S.~Verd\'{u}, ``Channel coding rate in the
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%  pp. 2307--2359, 2010.
%
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%V.~Y.~F. Tan, \emph{Asymptotic Estimates in Information Theory with
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%  Communications and Information Theory.\hskip 1em plus 0.5em minus 0.4em\relax
%  Now Publishers Inc, 2014, vol.~11.
%
%\bibitem{Hayashi09}
%M.~Hayashi, ``Information spectrum approach to second-order coding rate in
%  channel coding,'' \emph{IEEE Trans. Inf. Theory}, vol.~55, no.~11, pp.
%  4947--4966, 2009.
%
%\bibitem{TomTan12}
%M.~Tomamichel and V.~Y.~F. Tan, ``A tight upper bound for the third-order
%  asymptotics of most discrete memoryless channels,'' \emph{IEEE Trans. Inf.
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%
%\bibitem{elgamal}
%A.~{El~Gamal} and Y.-H. Kim, \emph{Network Information Theory}.\hskip 1em plus
%  0.5em minus 0.4em\relax Cambridge, U.K.: Cambridge University Press, 2012.
%
%\bibitem{Pol10}
%Y.~Polyanskiy, ``Channel coding: {Non}-asymptotic fundamental limits,'' Ph.D.
%  dissertation, Princeton University, 2010.
%
%\bibitem{TanTom13a}
%V.~Y.~F. Tan and M.~Tomamichel, ``The third-order term in the normal
%  approximation for the {AWGN} channel,'' \emph{{\tt arXiv:1311.2337 }}, Nov
%  2013.
%
%\bibitem{PPV11b}
%Y.~Polyanskiy, H.~V. Poor, and S.~Verd\'{u}, ``Feedback in the non-asymptotic
%  regime,'' \emph{IEEE Trans. Inf. Theory}, vol.~57, no.~8, pp. 4903--4925,
%  2011.
%
%\bibitem{feller}
%W.~Feller, \emph{An Introduction to Probability Theory and Its Applications},
%  2nd~ed.\hskip 1em plus 0.5em minus 0.4em\relax John Wiley and Sons, 1971.
%
%\bibitem{Wang09}
%L.~Wang, R.~Colbeck, and R.~Renner, ``Simple channel coding bounds,'' in
%  \emph{Proc. IEEE Intl. Symp. Inf. Theory}, Seoul, S. Korea, Jul. 2009, pp.
%  1804--1808.
%
%\bibitem{PPV10}
%Y.~Polyanskiy, H.~V. Poor, and S.~Verd\'{u}, ``Channel coding rate in the
%  finite blocklength regime,'' \emph{IEEE Trans. Inf. Theory}, vol.~56, no.~5,
%  pp. 2307--2359, 2010.
%
%\bibitem{TanBook}
%V.~Y.~F. Tan, \emph{Asymptotic Estimates in Information Theory with
%  Non-Vanishing Error Probabilities}, ser. Foundations and Trends in
%  Communications and Information Theory.\hskip 1em plus 0.5em minus 0.4em\relax
%  Now Publishers Inc, 2014, vol.~11.
%
%\bibitem{Hayashi09}
%M.~Hayashi, ``Information spectrum approach to second-order coding rate in
%  channel coding,'' \emph{IEEE Trans. Inf. Theory}, vol.~55, no.~11, pp.
%  4947--4966, 2009.
%
%\bibitem{TomTan12}
%M.~Tomamichel and V.~Y.~F. Tan, ``A tight upper bound for the third-order
%  asymptotics of most discrete memoryless channels,'' \emph{IEEE Trans. Inf.
%  Theory}, vol.~59, no.~11, pp. 7041--7051, 2013.
%
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{Sha49 C.~E. Shannon, ``Communication in the presence of noise,'' Proceedings of IRE, vol.~37, no.~1, pp. 10--21, 1949.
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1703.05932.extracted_bib
{Sha56 C.~E. Shannon, ``The zero error capacity of a noisy channel,'' IRE Transactions on Information Theory, vol.~2, no.~3, pp. 8--19, 1956.
1703.05932.extracted_bib
{AW14 Y.~Altu\u{g and A.~B. Wagner, ``Feedback can improve the second-order coding performance in discrete memoryless channels,'' in Proc. IEEE Intl. Symp. Inf. Theory, Honolulu, HI, Jul 2014, pp. 2361--2365.
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{PPV11b Y.~Polyanskiy, H.~V. Poor, and S.~Verd\'{u, ``Feedback in the non-asymptotic regime,'' {IEEE Trans. Inf. Theory, vol.~57, no.~8, pp. 4903--4925, 2011.
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{Machkouri M.~E. Machkouri and L.~Ouchti, ``Exact convergence rates in the central limit theorem for a class of martingales,'' Bernoulli, vol.~13, no.~4, pp. 981--999, Nov. 2007.
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{FongTan15 {S.~L.~Fong and V.~Y.~F.~Tan, ``{Asymptotic expansions for the {AWGN channel with feedback under a peak power constraint,'' in Proc. IEEE Intl. Symp. Inf. Theory, Hong Kong, Jun. 2015, pp. 311--315.
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{TFT17 L.~V. Truong, S.~L. Fong, and V.~Y.~F. Tan, ``On {Gaussian channels with feedback under expected power constraints and with non-vanishing error probabilities,'' {IEEE Trans. Inf. Theory, vol.~63, no.~3, pp. 1746--1765, 2017.
1703.05932.extracted_bib
{Wang09 L.~Wang, R.~Colbeck, and R.~Renner, ``Simple channel coding bounds,'' in Proc. IEEE Intl. Symp. Inf. Theory, Seoul, South Korea, June/July 2009, pp. 1804 -- 1808.
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{PPV10 Y.~Polyanskiy, H.~V. Poor, and S.~Verd\'{u, ``Channel coding rate in the finite blocklength regime,'' {IEEE Trans. Inf. Theory, vol.~56, no.~5, pp. 2307--2359, 2010.
1703.05932.extracted_bib
{curtiss J.~H. Curtiss, ``A note on the theory of moment generating functions,'' Ann. Math. Statist., vol.~13, no.~4, pp. 430--433, 1942.
1703.05932.extracted_bib
{Williams1991 D.~Williams, Probability with Martingales.\hskip 1em plus 0.5em minus 0.4em\relax Cambridge, U.K.: Cambridge University Press, 1991.
1703.05932.extracted_bib
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{Wang09 %L.~Wang, R.~Colbeck, and R.~Renner, ``Simple channel coding bounds,'' in % Proc. IEEE Intl. Symp. Inf. Theory, Seoul, S. Korea, Jul. 2009, pp. % 1804--1808. % %
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{PPV10 %Y.~Polyanskiy, H.~V. Poor, and S.~Verd\'{u, ``Channel coding rate in the % finite blocklength regime,'' IEEE Trans. Inf. Theory, vol.~56, no.~5, % pp. 2307--2359, 2010. % %
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{TanBook %V.~Y.~F. Tan, Asymptotic Estimates in Information Theory with % Non-Vanishing Error Probabilities, ser. Foundations and Trends in % Communications and Information Theory.\hskip 1em plus 0.5em minus 0.4em\relax % Now Publishers Inc, 2014, vol.~11. % %
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{Hayashi09 %M.~Hayashi, ``Information spectrum approach to second-order coding rate in % channel coding,'' IEEE Trans. Inf. Theory, vol.~55, no.~11, pp. % 4947--4966, 2009. % %
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{TomTan12 %M.~Tomamichel and V.~Y.~F. Tan, ``A tight upper bound for the third-order % asymptotics of most discrete memoryless channels,'' IEEE Trans. Inf. % Theory, vol.~59, no.~11, pp. 7041--7051, 2013. % %
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{CoverBook %{T.~M.~Cover and J.~A.~Thomas, {Elements of Information Theory, % 2nd~ed.\hskip 1em plus 0.5em minus 0.4em\relax Wiley-Interscience, 2006. % %
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{elgamal %A.~{El~Gamal and Y.-H. Kim, Network Information Theory.\hskip 1em plus % 0.5em minus 0.4em\relax Cambridge, U.K.: Cambridge University Press, 2012. % %
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{Pol10 %Y.~Polyanskiy, ``Channel coding: {Non-asymptotic fundamental limits,'' Ph.D. % dissertation, Princeton University, 2010. % %
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{TanTom13a %V.~Y.~F. Tan and M.~Tomamichel, ``The third-order term in the normal % approximation for the {AWGN channel,'' {\tt arXiv:1311.2337 , Nov % 2013. % %
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{PPV11b %Y.~Polyanskiy, H.~V. Poor, and S.~Verd\'{u, ``Feedback in the non-asymptotic % regime,'' IEEE Trans. Inf. Theory, vol.~57, no.~8, pp. 4903--4925, % 2011. % %
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{feller %W.~Feller, An Introduction to Probability Theory and Its Applications, % 2nd~ed.\hskip 1em plus 0.5em minus 0.4em\relax John Wiley and Sons, 1971. % %
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{Wang09 %L.~Wang, R.~Colbeck, and R.~Renner, ``Simple channel coding bounds,'' in % Proc. IEEE Intl. Symp. Inf. Theory, Seoul, S. Korea, Jul. 2009, pp. % 1804--1808. % %
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{PPV10 %Y.~Polyanskiy, H.~V. Poor, and S.~Verd\'{u, ``Channel coding rate in the % finite blocklength regime,'' IEEE Trans. Inf. Theory, vol.~56, no.~5, % pp. 2307--2359, 2010. % %
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{TanBook %V.~Y.~F. Tan, Asymptotic Estimates in Information Theory with % Non-Vanishing Error Probabilities, ser. Foundations and Trends in % Communications and Information Theory.\hskip 1em plus 0.5em minus 0.4em\relax % Now Publishers Inc, 2014, vol.~11. % %
1703.05932.extracted_bib
{Hayashi09 %M.~Hayashi, ``Information spectrum approach to second-order coding rate in % channel coding,'' IEEE Trans. Inf. Theory, vol.~55, no.~11, pp. % 4947--4966, 2009. % %
1703.05932.extracted_bib
{TomTan12 %M.~Tomamichel and V.~Y.~F. Tan, ``A tight upper bound for the third-order % asymptotics of most discrete memoryless channels,'' IEEE Trans. Inf. % Theory, vol.~59, no.~11, pp. 7041--7051, 2013. % %