Weakly interacting particle systems on inhomogeneous random graphs
Shankar Bhamidi
Amarjit Budhiraja and Ruoyu Wu
\
We consider weakly interacting diffusions on time varying random graphs. The system consists of a large number of nodes in which the state of each node is governed by a diffusion process that is influenced by the neighboring nodes. The collection of neighbors of a given node changes dynamically over time and is determined through a time evolving random graph process. A law of large numbers and a propagation of chaos result is established for a multi-type population setting where at each instant the interaction between nodes is given by an inhomogeneous random graph which may change over time. This result covers the setting in which the edge probabilities between any two nodes is allowed to decay to $0$ as the size of the system grows. A central limit theorem is established for the single-type population case under stronger conditions on the edge probability function.
1612.00801.tex
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% \author{Shankar Bhamidi, Amarjit Budhiraja and Ruoyu Wu \\ \ \\}
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% %\noi {\bf AMS 2000 subject classifications:} .
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% %\noi {\bf Keywords:} Mean field interaction, common factor, weakly interacting diffusions, propagation of chaos, central limit theorems, fluctuation limits, symmetric statistics, multiple Wiener integrals.
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\begin{document}


\title[WIPS on random graphs]{Weakly interacting particle systems on inhomogeneous random graphs}

\date{}
\subjclass[2010]{Primary: 60C05, 05C80, 60F05, 60K35, 60H30, 60J70. }% Secondary: ;}
\keywords{inhomogeneous random graphs, dynamical random graphs, weakly interacting diffusions, propagation of chaos, central limit theorems, multi-type populations, interacting particle systems. }

\author[Bhamidi]{Shankar Bhamidi$^1$}
\address{$^1$Department of Statistics and Operations Research, 304 Hanes Hall, University of North Carolina, Chapel Hill, NC 27599}
\author[Budhiraja]{Amarjit Budhiraja$^1$}
\author[Wu]{Ruoyu Wu$^2$}
\address{$^2$Division of Applied Mathematics, 182 George Street, Brown University, Providence, RI 02912}
\email{bhamidi@email.unc.edu,  budhiraj@email.unc.edu, ruoyu\_wu@brown.edu}
\maketitle
\begin{abstract}
	We consider weakly interacting diffusions on time varying random graphs. The system consists of a large number of nodes in which the state of each node is governed by
	a diffusion process that is influenced by the neighboring nodes. The collection of neighbors of a given node changes dynamically over time and is determined through a time evolving random graph process.
	A law of large numbers and a propagation of chaos result is established for a multi-type population setting where at each instant the interaction between nodes is given by an inhomogeneous random graph
	which may change over time. This result covers the setting in which the edge probabilities between any two nodes is allowed to decay to $0$ as the size of the system grows. A central limit theorem is established for the single-type population case under stronger conditions on the edge probability function.
\end{abstract}



%----------------------------------------------------------

\section{Introduction} \label{sec:intro}

In this work we study some asymptotic results for large particle systems given as weakly interacting diffusion processes on time varying inhomogeneous random graphs. 
The model is described in terms of two types of stochastic dynamical systems, one that describes the evolution of 
the  graph that governs the interaction between nodes of the system ({\em dynamics of the network}) and the other that describes the evolution of the states of all the nodes in the system ({\em dynamics on the network}). 
We consider a setting where the interaction between the nodes is {\em weak} in that the `strength' of the interaction between a node and its neighbor is inversely proportional to the total number of neighbors of that node. 
Such stochastic systems arise in many different areas, such as social networks (e.g.\ in the study of gossip algorithms \cite{bertsekas1989parallel,shah2009gossip}), biological systems (e.g.\ swarming and flocking models, see \cite{bertozzi1} and references therein), neurosciences (e.g.\ in modeling of networks of spiking neurons, see \cite{BaladronFaugeras2012,Bossy2015clarification} and references therein), and mathematical finance (e.g.\ in modeling correlations between default probabilities of multiple firms \cite{CvitanicMaZhang2012law}). 

The case where the interaction graph is complete, i.e.\ each node interacts with every other node, is classical and dates back to works of Boltzmann, Vlasov, McKean and others (see \cite{Sznitman1991,Kolokoltsov2010} and references therein). Original motivation for the study of such systems came from Statistical Physics but, as noted above, in recent years similar models have arisen in many different application areas, ranging from mathematical finance and chemical and biological systems to communication networks and social sciences (see \cite{BudhirajaDupuisFischerRamanan2015limits} for an extensive list of references).
The asymptotic picture in the setting of a complete graph is well resolved and many different results have been established, including laws of large numbers (LLN), propagation of chaos (POC) properties, and central limit theorems (CLT), see e.g.\ \cite{McKean1966class,McKean1967propagation,BraunHepp1977vlasov,Dawson1983critical,Tanaka1984limit,Oelschlager1984martingale,Sznitman1984,Sznitman1991,GrahamMeleard1997,ShigaTanaka1985,Meleard1998}.
 % Braun and Hepp~\cite{BraunHepp1977vlasov}, Dawson~\cite{Dawson1983critical}, Tanaka~\cite{Tanaka1984limit}, Oelschal\"{a}ger~\cite{Oelschlager1984martingale}, Sznitman~\cite{Sznitman1984,Sznitman1991}, Graham and M\'{e}l\'{e}ard~\cite{GrahamMeleard1997}, Shiga and Tanaka~\cite{ShigaTanaka1985}, M\'{e}l\'{e}ard~\cite{Meleard1998}.
 A number of variations have also been studied.
For example, in~\cite{KurtzXiong1999,KurtzXiong2004,BudhirajaSaha2014,BudhirajaWu2015} a setting where a common noise process  influences the dynamics of every particle is considered and limit theorems of the above form are established.
For a setting with $K$-different subpopulations within each of which particle evolution is exchangeable, LLN and POC have been studied in~\cite{BaladronFaugeras2012}, and a corresponding CLT has been established in~\cite{BudhirajaWu2015}.
Mean field results for heterogeneous populations have also been  studied in ~\cite{Collet2014macroscopic,ChongKluppelberg2015partial}. All of these  papers consider  complete interaction graphs where every vertex influences every other vertex (albeit in a `weak' fashion).

\subsection{Informal overview of our contributions and proof techniques}
\label{sec:inf-over}

The goal of the current work is to develop an analogous limit theory when the interaction graph is not complete and is possibly time varying. 
The two main results of this work are Theorem \ref{thm:NpN_rate}
and Theorem \ref{thm:CLT}.  Theorem \ref{thm:NpN_rate} and its corollaries (Corollary \ref{cor:poc} and Corollary \ref{cor:LLN}) give a law of large numbers result and a propagation of chaos property whereas Theorem
\ref{thm:CLT} proves a central limit theorem.


 In the area of interacting particle systems there is a large amount of work for many different settings with dynamics on random graphs, for example, the voter model or the contact process \cite{aldous2013interacting,liggett2012interacting,durrett1995ten,durrett-book}. There is much less work for the class of models considered in this paper. The closest in spirit to the work here is \cite{bertozzi1,bertozzi2} where the authors consider an \erdos random graph $\Gmc_N(p) = \set{\xi_{ij}: 1\le i< j \le N }$, where $\xi_{ij}=1$ if there is an edge between vertices $i$ and $j$, and $0$ otherwise.  Using the interaction structure generated by this graph these papers consider a  family of coupled ordinary differential equations (ODE),
\[\frac{d z_i}{dt} = \frac{1}{N} \sum_{j=1}^N \xi_{ij} F(|z_i - z_j|) \frac{z_i - z_j}{|z_i - z_j|}, \qquad i=1,\ldots, N. \]
Here $F(\cdot)$ is a suitable repulsive-attractive force that captures the phenomenon that particles are attracted to each other unless they get too close, in which case  they are repelled.
The aim of these works is  to understand the consensus behavior of the associated coupled system of ODE as $N\to\infty$. The model considered in the current work is different in several aspects: (1) The random graph is allowed to change over time in quite a general fashion; (2) the population can be multi-type, namely the associated random graph can be `inhomogeneous'; (3) Edge probabilities can change with system size and are even allowed to decay to zero (for the LLN result) as system size increases; (4) the dynamics of the node states can have external noise and be described through stochastic differential equations (SDE) rather than ODE. 
Another recent work that is close to ours is \cite{Delattre2016} where interacting diffusions on static graphs are studied and quenched results on propagation of chaos are established.



For the law of large numbers result, we consider a random graph model that is a time evolving version of the inhomogeneous random graph models studied by Bollobas, Janson and Riordan in \cite{bollobas-riordan-janson}.
Roughly speaking, each node can be of $K$ possible types and at any time instant edges between two nodes form independently with probabilities depending only on the types of the two nodes.
These probabilities may change over time and are allowed to decay with $N$ (see Condition \ref{cond:cond1}). The evolution of the node states is described through a collection of weakly interacting stochastic
differential equations(SDE) such that the interaction of a particular particle with its neighbors is given through the coefficients in the SDE for that particle that depend on the states of all its neighbors through functions that depend only on the node-type (see equation \eqref{eq:ZiNt}
for a precise description of the evolution). The proof of the law of large numbers relies on certain coupling arguments along with exchangeability properties of nodes of each type, and concentration inequalities for various functionals of the random graph.

Our second main result proves a central limit theorem for scaled functionals of the empirical measures of the particle states. For simplicity, here we consider the single type setting, namely $K=1$. 
The proof of CLT relies on a change of measure technique using Girsanov's theorem, which goes back to \cite{Sznitman1984,ShigaTanaka1985}.
This technique reduces the problem to a setting with i.i.d.\ particles and edges, while the price to pay is that one must carefully analyze the asymptotic behavior of the Radon-Nikodym derivative $J^N(T)$, which is given in terms of quantities (see $J^{N,1}(T)$ and $J^{N,2}(T)$ in \eqref{eq:JN1} and \eqref{eq:JN2} for the martingale and the quadratic variation part, respectively) involving all particles and edges.
%Due to the presence of random interaction, the Radon-Nikodym derivative is  
%which cannot be viewed as a collection of i.i.d.\ particle-edge combinations.
For the asymptotics of the quadratic variation part $J^{N,2}(T)$, one can modify the system to reduce the analysis to the setting of a complete graph by carefully studying the effect of the time evolving random interaction graph (Lemma \ref{lem:Ttil}) and bounding the corresponding error (Lemma \ref{lem:tiljn2}), and then apply classical limit theorems for symmetric statistics \cite{Dynkin1983}.
For the martingale part $J^{N,1}(T)$, however, one cannot easily modify the system to replace the random graph  by a complete graph while keeping the  particle interaction to be  i.i.d.
Due to the interaction being governed by a time evolving random graph, neither can one  apply classical techniques from \cite{Sznitman1984,ShigaTanaka1985}, since the entire collection of particles and edges cannot be viewed as a collection of i.i.d.\ particle-edge combinations. 
The asymptotic analysis of $J^{N,1}(T)$ presents one of the main challenges in the proof.
%Hence one cannot directly apply the classical results on asymptotics for symmetric statistics \cite{Dynkin1983} that are used in \cite{Sznitman1984,ShigaTanaka1985} and the main challenge in this work is to study asymptotics for these quantities caused by the random interaction.
For this, we first show in Lemma \ref{lem:JN1_JN1_tilde} that $J^{N,1}(T)$ is asymptotically close to $\Jtil^{N,1}(T)$ defined in \eqref{eq:JN1til}, which is written in a form similar to an incomplete U-statistic and that has `lesser dependence' on the random interaction graph. The significance of this result is that incomplete U-statistics, first introduced by Blom \cite{Blom1976some}, have been shown to be asymptotically normal by  Janson \cite{Janson1984}, under suitable conditions. We extend Janson's limit result to the setting of stochastic processes where the incompleteness shows up in the integrands of certain stochastic integrals  and combine it with classical limit theorems for symmetric statistics, to obtain  in Lemma \ref{lem:key}, a key characterization of the asymptotic distribution of $\Jtil^{N,1}(T)$ as the sum of a normal random variable and a multiple Wiener integral (see Section \ref{sec:asymp_symmetric_statistics} for precise definitions) with certain independence properties.
These properties allow us to obtain joint asymptotic behavior of $J^{N,1}(T)$ and $J^{N,2}(T)$, which completes the analysis of the Radon-Nikodym derivative.

The proof of the central limit theorem requires a stronger condition on the edge probabilities (see Condition \ref{cond:cond2}) than that needed for the law of large numbers. In particular, here we are unable to treat the case where the edge probabilities decay to $0$ with $N$.
One interesting aspect of the proof is the role of the non-degeneracy assumption on edge probabilities in Condition \ref{cond:cond2}. For simplicity, consider the setting where the random graph is static, given at time instant $0$ according to an \Erdos random graph with edge probabilities $p_N$ and suppose that $p_N \to p>0$ as $N \to \infty$. Then, as seen in Theorem \ref{thm:CLT}, the variance of the limiting Gaussian random field does not depend on $p$. One may conjecture that because of this fact one may be able to relax the condition $p>0$ and allow $p_N$ to converge to $0$ at an appropriate rate. However, as noted earlier, a crucial ingredient in our proof is the study of the asymptotic behavior of the Radon-Nikodym derivative $J^N(T)$. The limit of this random variable is described (see Proposition \ref{prop:key_joint_cvg})
in terms of a Normal random variable $Z$ (along with a multiple Wiener integral of order $2$) with mean $b(p)$ and variance $\sigma^2(p)$ where both $b(p)$ and $\sigma^2(p)$ approach $\infty$ as $p \to 0$.
The arguments of Section \ref{sec:complete_pf_CLT} show that the distribution of the limiting Gaussian random field depend on the quantity $\Ebf e^Z = e^{b(p)+ \sigma^2(p)/2}$ and one finds that although both
$b(p)$ and $\sigma^2(p)$ diverge, the quantity $b(p)+ \sigma^2(p)/2 =0$ for each $p>0$. This is the key observation in the proof and the reason the limit random field does not depend on $p$ . However, when $p_N \to 0$, a similar analysis of the asymptotics of the Radon-Nikodym derivative $J^N(T)$ cannot be carried out and even the tightness of this term is unclear. Proving a suitable fluctuations result in this regime (i.e.\ when $p_N \to 0$ at a suitable rate) is an interesting and challenging open problem.


% In the context of applications in areas such as robotics or flocking models in mathematical biology, for the importance of considering random interaction graphs with possibly dynamic evolution of the network structure, see e.g.\ \cite{aldana2007phase,huepe2011adaptive,porfiri2006random,porfiri2007consensus}. 


\subsection{Organization}
\label{sec:org}


The paper is organized as follows. We conclude this Section by outlining the notation used in the rest of the paper. 
Our model of weakly interacting multi-type diffusions on random graphs is introduced in Section \ref{sec:mod}. % Section \ref{sec:not} collects some common notation used in this work.
In Section \ref{sec:laws-large-results} two basic conditions (Conditions \ref{cond:coefficients} and \ref{cond:cond1}) on coefficients in the model and sparsity of the interaction graph are stated, under which a law of large numbers and propagation of chaos property are established in Theorem \ref{thm:NpN_rate} and its corollaries.
Next in Section \ref{sec:fluctuations} we present a central limit theorem (Theorem \ref{thm:CLT}) in the single-type setting  under a stronger condition (Condition \ref{cond:cond2}) on sparsity of the interaction graph.
%In Section \ref{sec:disc} the connection between our model and related literature are discussed.
The rest of this paper gives proofs of Theorems \ref{thm:NpN_rate} and \ref{thm:CLT}.
We start in Section \ref{sec:pre_results} with some preliminary concentration results on the degree distribution.
Theorem \ref{thm:NpN_rate} is proved in Section \ref{sec:pf_NpN_rate}.
In Section \ref{sec:pf_CLT} we give the proof of Theorem \ref{thm:CLT}.
The proofs of several technical Lemmas and auxiliary results are given in the Appendix.

\subsection{Notation}
\label{sec:not}
The following notation will be used in the sequel. 
For a Polish space $(\Smb, d(\cdot,\cdot))$, denote the corresponding Borel $\sigma$-field by $\Bmc(\Smb)$.
For a signed measure $\mu$ on $\Smb$ and $\mu$-integrable function $f \colon \Smb \to \Rmb$, let $\langle f,\mu \rangle \doteq \int f \, d\mu$.
Denote by $\Pmc(\Smb)$ (resp.\ $\Mmc(\Smb)$) the space of probability measures (resp.\ sub-probability measures) on $\Smb$, equipped with the topology of weak convergence.
A convenient metric for this topology is the bounded-Lipschitz metric $d_{BL}$, defined as
\begin{equation*}
d_{BL}(\nu_1,\nu_2) \doteq \sup_{\|f\|_{BL} \le 1} | \langle f, \nu_1 - \nu_2 \rangle |, \quad \nu_1, \nu_2 \in \Mmc(\Smb),
\end{equation*}
where $\|\cdot\|_{BL}$ is the bounded Lipschitz norm, i.e.\ for $f \colon \Smb \to \Rmb$,
\begin{equation*}
	\|f\|_{BL} \doteq \max \{\|f\|_\infty, \|f\|_L\}, \quad \|f\|_\infty \doteq \sup_{x \in \Smb} |f(x)|, \quad \|f\|_L \doteq \sup_{x \ne y} \frac{|f(x)-f(y)|}{d(x,y)}.
\end{equation*}
%where $d$ is the metric on $\Smb$.
Denote by $\Cmb_b(\Smb)$ the space of real bounded and continuous functions.
%For Banach spaces $B_1$ and $B_2$, $L(B_1,B_2)$ will denote the space of bounded linear operators from $B_1$ to $B_2$.
For a measure $\nu$ on $\Smb$ and a Hilbert space $H$, let $L^2(\Smb,\nu,H)$ denote the space of measurable functions $f \colon \Smb \to H$ such that $\int_\Smb \| f(x) \|^2_H \, \nu(dx) < \infty$, where $\| \cdot \|_H$ is the norm on $H$.
When $H = \Rmb$, we write $L^2(\Smb,\nu)$ or simply $L^2(\nu)$ if  $\Smb$ is clear from context.
%The norm in this Hilbert space will be denoted as $\| \cdot \|_{L^2(\Smb,\nu)}$.
%Let $\Cmb_b^k(\Rmb^d)$ be the space of functions on $\Rmb^d$, which have continuous and bounded partial derivatives up to the $k$-th order.

Fix $T < \infty$. 
All stochastic processes will be considered over the time horizon $[0,T]$. 
We will use the notations $\{X_t\}$ and $\{X(t)\}$ interchangeably for stochastic processes.
For a Polish space $\Smb$, denote by $\Cmb([0,T]:\Smb)$ (resp.\ $\Dmb([0,T]:\Smb)$) the space of continuous functions (resp.\ right continuous functions with left limits) from $[0,T]$ to $\Smb$, endowed with the uniform topology (resp.\ Skorokhod topology). 
%When $\Smb$ is a normed space with norm $\| \cdot \|$, for a map $f \colon [0,T] \to \Smb$, let $\|f\|_{*,t} \doteq \sup_{0 \le s \le t} \|f(s)\|$, $t \in [0,T]$.
For $d \in \Nmb$, let $\Cmc_d \doteq \Cmb([0,T]:\Rmb^d)$ and $\|f\|_{*,t} \doteq \sup_{0 \le s \le t} \|f(s)\|$ for $f \in \Cmc_d$, $t \in [0,T]$.
We say a collection $\{ X^m \}$ of $\Smb$-valued random variables is tight if the distributions of $X^m$ are tight in $\Pmc(\Smb)$.
We use the symbol `$\Rightarrow$' to denote convergence in distribution.
The distribution of an $\Smb$-valued random variable $X$ will be denoted as $\Lmc(X)$. 
Expected value under a probability distribution $\Pmb$ will be denoted as $\Ebf_{\Pmb}$ but when clear from the context,
$\Pmb$ will be suppressed from the notation.

%For a $\sigma$-finite measure $\nu$ on a Polish space $\Smb$, denote by $L^2_{\Rmb^k}(\Smb,\nu)$ as the Hilbert space of $\nu$-square integrable functions from $\Smb$ to $\Rmb^k$. 
%When $k$ and $\Smb$ are not ambiguous, we will merely write $L^2(\nu)$. 
%The norm in this Hilbert space will be denoted as $\| \cdot \|_{L^2(\Smb,\nu)}$.
%Given a sequence of random variables $X_n$ on some probability space $(\Omega_n, \Fmc_n, P_n)$, $n \ge 1$, we say $X_n$ converges to $0$ in $L^2(\Omega_n, P_n)$ if $\int X_n^2 \, dP_n \to 0$ as $n \to \infty$.
%We will usually denote by $\kappa, \kappa_1, \kappa_2, \dotsc$, the constants that appear in various estimates within a proof. 
%The value of these constants may change from one proof to another.
%We use $\Nmb_0$ to denote the set of non-negative integers.
%Cardinality of a finite set $A$ will be denoted as $|A|$.
%$B_r^d$ denotes the closed ball of radius $r$ in $\Rmb^d$, centered at origin, i.e.\ $B_r^d = \{ x \in \Rmb^d : \| x \| \le r \}$.
%For $x,y \in \Rmb^d$, let $x \cdot y = \sum_{i=1}^d x_i y_i$.






%Given a sequence of random variables $X_n$ on probability spaces $(\Omega_n, \Fmc_n, P_n)$, $n \ge 1$, we say $X_n$ converges to $0$ in $L^2(\Omega_n, P_n)$ if $\int X_n^2 \, dP_n \to 0$ as $n \to \infty$.
We will usually denote by $\kappa, \kappa_1, \kappa_2, \dotsc$, the constants that appear in various estimates within a proof. 
The value of these constants may change from one proof to another.
%Cardinality of a finite set $A$ will be denoted as $|A|$.
%For $x,y \in \Rmb^d$, let $x \cdot y \doteq \sum_{i=1}^d x_i y_i$.






\section{Model}
\label{sec:mod}

We now describe the precise model that will be studied in this work. Law of large numbers will be established in the general $K$-type setting described below whereas  for the central limit theorem in Section \ref{sec:fluctuations} we will consider for simplicity the case $K=1$.
 
\subsection{Random graph}
Our random graph model is a (possibly time evolving) version in the class of inhomogeneous random graph models studied by Bollobas, Janson and Riordan in \cite{bollobas-riordan-janson}. 
We start with $N$ vertices represented by the vertex set $\Nbd \doteq \set{1,2,\ldots, N}$. 
Assume that each vertex can be one of $K$ possible types (sometimes called populations) labelled according to $\Kbd \doteq \set{1,2,\ldots, K}$. We will require, for every $\alpha \in  \Kbd$, the number of type $\alpha$ vertices, denoted by $N_{\alpha}$, to approach infinity and the ratio $N_{\alpha}/N$ to converge to  a  positive value as $N\to \infty$.
It will be notationally convenient to assume that $N_{\alpha}$ are nondecreasing in $N$ so that by reindexing if needed the type $\alpha$ vertices in the $N$-th model can be described through a membership map $\pbd: \Nmb \to \Kbd$ such that
type $\alpha$-vertices in the $N$-th model are given as $\Nalpha \doteq \{i \in \Nbd: \pbd(i)=\alpha\}$.
%For any $\alpha \in \Kbd$ write $\Nalpha$ for the collection of all vertices of type $\alpha$. 
In this work we take the membership map to be fixed over time. It may be viewed as a given deterministic function or determined through a sample realization (which is then fixed throughout) of $N$ i.i.d.\ 
$\Kbd$-valued random variables
$\{p(i)\}_{i=1}^N$ distributed according to some probability distribution ${\boldsymbol\pi} = (\pi(\alpha), \alpha \in \Kbd)$ on $\Kbd$.  The evolution of the random graph is given in terms of a stochastic process
$\set{\xi_{ij}^N(t): t\in [0,T], 1\leq i\leq j\leq N}$ given on some filtered probability space $(\Omega,\Fmc,\Pbd,\{\Fmc_t\})$. We assume that 
% 
% 
% 
% 
% 
% One needs to prescribe how the types of various individuals are chosen: one natural assumption is that for each $N\geq 1$ we are given a probability vector ${\boldsymbol\pi}_N = (\pi_N(\alpha), \alpha \in \Kbd)$. 
% Now every vertex chooses its type in an i.i.d.\ fashion using ${\boldsymbol\pi}_N$, namely $\Pbd(\pbd(i) = \alpha) = \pi_N(\alpha)$, $\alpha \in \Kbd$. 
% 
% Now assume we are given a filtered probability space $(\Omega,\Fmc,\Pbd,\{\Fmc_t\})$. 
% Using this space, we assume we can construct a time evolving random graph $\set{\xi_{ij}^N(t): t\geq 0, 1\leq i\leq j\leq N}$ on these $N$ vertices.  
$\xi_{ii}^N(t) \equiv 1$ for all $t\in [0,T]$ while $\{\xi_{ij}^N(t) \equiv \xi_{ji}^N(t): 1 \le i < j \le N\}$ are mutually independent $\{\Fmc_t\}$-adapted RCLL(right continuous with left limits) processes such that
\begin{equation*}
	\Pbd(\xi_{ij}^N(t)=1) = 1 - \Pbd(\xi_{ij}^N(t)=0) = p_{\alpha\gamma,N}(t), \quad \alpha, \gamma \in \Kbd, i \in \Nalpha, j \in \Ngamma, i \ne j,
\end{equation*}
where $p_{\alpha\gamma,N}$ is a $[0,1]$-valued continuous function on $[0,T]$ for every $\alpha, \gamma \in \Kbd$, $N \in \mathbb{N}$.
Note that $p_{\alpha\gamma,N}=p_{\gamma\alpha,N}$ for all $\alpha,\gamma \in \Kbd$.
Given such a stochastic process, for any $t\in [0,T]$  the graph $\Gmc_N(t)$ is formed via the following procedure:  for any unordered pair $i\neq j \in \Nbd$ if $\xi_{ij}^N(t) =1$, this implies at that given time instant $t$, there exists an edge between vertex $i$ and $j$; if $\xi_{ij}^N(t) =0$ then no edge exists at time $t$. The self-loop edges $\xi_{ii}^N(\cdot) \equiv 1$ are used just as a simplification in the interacting particle system process described in the next subsection. 
Ignoring these self-loop edges, the random graph $\Gmc_N(t)$ belongs to the class of inhomogeneous random graph models analyzed in \cite{bollobas-riordan-janson} wherein edges are formed randomly between vertices and connection probabilities depend only on the `type' of the vertex. For later use, let $N_{i,\gamma}(t) \doteq \sum_{j \in \Ngamma} \xi_{ij}^N(t)$ for $i \in \Nbd$, $\gamma \in \Kbd$. 
Thus $N_{i,\gamma}(t)$ represents the number of $\gamma$-th type neighbors of vertex $i$ at time $t$. 
Note that $N_{i,\alpha}(t) \ge 1$ for all $t \in [0,T]$ and $i \in \Nalpha$. 
Define,
\begin{equation}
	\label{eqn:def-pbar}
	\pbar_N\doteq \inf_{t \in [0,T]} \min_{\alpha,\gamma \in \Kbd} p_{\alpha\gamma,N}(t).
\end{equation}
%When there is no scope for confusion we will suppress dependence on $T$ and write the above as $\pbar_N:=\pbar_N(T)$. 
The following are two natural families for the edge process. 

\begin{example}[{\bf Static networks}] \label{eg:eg1}
	Let $\{\xi_{ij}^N(t)\}$ be unchanging over time, namely $\xi_{ij}^N(t) \equiv \xi_{ij}^N(0)$.
	In this case 
	\begin{equation*}
		\pbar_N = \min_{\alpha,\gamma \in \Kbd} p_{\alpha\gamma,N}(0), \quad N_{i,\gamma}(t) \equiv N_{i,\gamma}(0)
	\end{equation*} 
	for all $t \in [0,T]$, $i,j \in \Nbd$ and $\gamma \in \Kbd$. 
\end{example}

\begin{example}[{\bf Markovian  edge formation}] \label{eg:eg2}
For each $N\geq 1$ let $\{\xi^N_{ij}(0)\}_{1\le i < j \le N}$ be mutually independent $\{0,1\}$-valued random variables
such that $P( \xi_{ij}^N(0) = 1 ) = p_{\alpha\gamma,N}(0)$ for $i\neq j\in \Nbd$ with $i\in \Nalpha$ and $j\in \Ngamma$
and $\alpha, \gamma \in \Kbd$. Fix positive $\{\lambda_{\alpha\gamma,N} = \lambda_{\gamma\alpha,N}\}_{\alpha,\gamma \in \Kbd}$ and $\{\mu_{\alpha\gamma,N} = \mu_{\gamma\alpha,N}\}_{\alpha,\gamma \in \Kbd}$.	
For any two vertices $i\neq j\in \Nbd$ with $i\in \Nalpha$ and $j\in \Ngamma$, let 
%$\{\xi_{ij}^N(t)\}$ change values independently according to exponential clocks modulated using these constants, namely 
$\set{\xi_{ij}^N(t): t\geq 0}$ be a $\{0,1\}$-valued Markov process with rate matrix,
	\begin{equation*}
		\Gamma_{\alpha\gamma,N} \doteq 
		\begin{bmatrix}
			-\lambda_{\alpha\gamma,N}	& \lambda_{\alpha\gamma,N} \\
			\mu_{\alpha\gamma,N}		& -\mu_{\alpha\gamma,N}
		\end{bmatrix}.
	\end{equation*}
	We assume that the evolutions of Markov chains for different edges are independent. In this setting
	\begin{equation*}
		p_{\alpha\gamma,N}(t) = p_{\alpha\gamma,N}(0) e^{-(\lambda_{\alpha\gamma,N}+\mu_{\alpha\gamma,N})t} + \frac{\lambda_{\alpha\gamma,N}}{\lambda_{\alpha\gamma,N} + \mu_{\alpha\gamma,N}} \left( 1-e^{-(\lambda_{\alpha\gamma,N}+\mu_{\alpha\gamma,N})t} \right)
	\end{equation*}
	and hence 
	\begin{equation*}
		\pushQED{\qed} \pbar_N = \min_{\alpha,\gamma \in \Kbd} \min \{ p_{\alpha\gamma,N}(0), p_{\alpha\gamma,N}(T) \} \ge \min_{\alpha,\gamma \in \Kbd} \min \Big\{ p_{\alpha\gamma,N}(0), \frac{\lambda_{\alpha\gamma,N}}{\lambda_{\alpha\gamma,N} + \mu_{\alpha\gamma,N}} \Big\}. 
		%\qedhere \popQED
	\end{equation*}
\end{example}
One can also allow non-Markovian edge formation processes where the holding times have general probability distributions that
satisfy appropriate conditions.



\subsection{Interacting particle system}
The main object of interest in this paper is a collection of $\Rd$-valued diffusion processes $\{Z^{1,N},\dotsc,Z^{N,N}\}$, representing trajectories of $N$  particles of $K$ types and that interact through the evolving graphical structure represented by $\set{\Gmc_N(t):t\geq 0}$. 
The dynamics is given in terms of a collection of stochastic differential equations (SDE) driven by mutually independent Brownian motions with each particle's initial condition governed independently by a probability law that depends only on its type.
The interaction between particles occurs through the coefficients of the SDE in that for the $i$-th particle $Z^{i,N}$, with $\pbd(i)=\alpha$, the coefficients depend on not only the $i$-th particle's current state, but also $K$ empirical measures of its neighbors corresponding to $K$ types. 
A precise formulation is as follows. Recall the filtered probability space $(\Omega,\Fmc,\Pbd,\{\Fmc_t\})$ on which the edge processes $\set{\xi_{ij}^N: i\leq j \in \Nbd}$ are given. 
We suppose that on this space we are also given an infinite collection of standard $d$-dimensional $\{\Fmc_t\}$-Brownian motions $\{W^i : i \in \Nbd\}$ and $\Fmc_0$-measurable $\Rd$-valued random variables $\{X^i_0 : i \in \Nbd \}$, with $\Lmc(X^i_0) = \mu^\alpha_0$ for $i \in \Nalpha, \alpha \in \Kbd$, such that  $\{W^i, X^i_0, \xi_{ij}^N: i\leq j \in \Nbd\}$ are mutually independent.

% \[\mu_t^{i,\gamma,N} \doteq \frac{1}{N_{i,\gamma}(t)} \sum_{j:\pbd(j)=\gamma} \xi_{ij}^N(t) \delta_{Z_t^{j,N}} \one_{\{N_{i,\gamma}(t) > 0\}}, \qquad \gamma \in \{1,\dotsc,K\}, \qquad i\in \Nbd. \]
Recall that $N_{i,\gamma}(t) \doteq \sum_{j \in \Nbd_\gamma} \xi_{ij}^N(t)$ denotes the number of $\gamma$-th type particles that interact with $i$-th particle at time instant $t$. For $\alpha \in \Kbd$ and $i \in \Nalpha$ consider the collection of stochastic differential equations given by,
\begin{align}
	\ZiNt & = X^i_0 + \sum_{\gamma=1}^K \int_0^t b_{\alpha\gamma}(\ZiNs,\mu^{i,\gamma,N}_s) \, ds + \sum_{\gamma=1}^K \int_0^t \sigma_{\alpha\gamma}(\ZiNs,\mu^{i,\gamma,N}_s) \, dW^i_s, \label{eq:ZiNt} \\
	\mu^{i,\gamma,N}_t & = \frac{1}{N_{i,\gamma}(t)} \sum_{j \in \Ngamma} \xi_{ij}^N(t) \delta_{\ZjNt} \one_{\{N_{i,\gamma}(t) > 0\}},  
\end{align}
where for $\alpha,\gamma \in \Kbd$, $x \in \Rmb^d$ and $\theta \in \Mmc(\Rd)$, 
%(the space of sub-probability measures on $\Rd$ endowed with the usual weak topology), 
%(space of sub-probability measures on $\Rd$, equipped with the usual topology of weak convergence), 
\begin{equation*}
	b_{\alpha\gamma}(x,\theta) \doteq \intRd \bbar_{\alpha\gamma}(x,y) \, \theta(dy), \quad \sigma_{\alpha\gamma}(x,\theta) \doteq \intRd \sigmabar_{\alpha\gamma}(x,y) \, \theta(dy)
\end{equation*} 
for suitable functions $\bbar_{\alpha\gamma} \colon \Rd \times \Rd \to \Rd$ and $\sigmabar_{\alpha\gamma} \colon \Rd \times \Rd \to \Rmb^{d \times d}$.
%$\{W^i: i \in \Nbd \}$ are i.i.d.\ $d$-dimensional $\{\Fmc_t\}$-Brownian motions. 
Under Condition \ref{cond:coefficients} one can easily establish  existence and uniqueness of pathwise solutions to the above system of stochastic differential equations  \cite{Sznitman1991}.

We can now summarize the main contributions of this work.
\begin{enumeratei}
	\item In Theorem \ref{thm:NpN_rate} and its corollaries we show that with suitable assumptions on coefficients (Condition \ref{cond:coefficients}) and a sparsity condition on the interaction graph that is formulated in terms of the decay rate of $\pbar_N$ (Condition \ref{cond:cond1}), a law of large numbers and propagation of chaos result hold.
\item In Section \ref{sec:fluctuations} we study the fluctuations of $\{Z^{i,N}\}$ from its law of large numbers limit by establishing a central limit theorem.
For simplicity, we study the single-type setting, i.e.\ $K=1$. 
%where the summands are functionals of the trajectories of the particles with suitable integrability properties.
Specifically, let
\begin{equation*}
	\eta^N(\phi) \doteq \frac{1}{\sqrt{N}} \sum_{i=1}^N \phi(Z^{i,N}), \quad \phi \in L^2_c(\Cmc_d,\mu),
\end{equation*}
where $L^2_c(\Cmc_d,\mu)$ is a family of functions on the path space that are suitably centered and have appropriate integrability properties (see Section \ref{sec:CLT} for definitions).
We show in Theorem \ref{thm:CLT} that under Condition \ref{cond:coefficients} and a stronger assumption on edge probability $p_N$ (Condition \ref{cond:cond2}) the family $\{\eta^N(\phi) : \phi \in L^2_c(\Cmc_d,\mu) \}$ converges weakly to a mean $0$ Gaussian field $\{\eta(\phi) : \phi \in L^2_c(\Cmc_d,\mu) \}$ in the sense of convergence of finite dimensional distributions.
% The variance-covariance structure of this Gaussian field is the same as that in the setting where the interaction graph is complete, which indicates that as long as the interaction graph is not too sparse (see Condition \ref{cond:cond2}), the asymptotic behavior of such general system is similar to the one with a complete interaction graph.
\end{enumeratei}


% ********************
% Over the last few years there has been an explosion in the amount of data on a host of real world networks ranging from social and epidemic contact networks, information networks such as the Internet to biochemical networks such as gene regulatory networks. This has caused an explosion in a number of fields in developing models to explain functionals observed in these networks as well as gain insight into various dynamic processes on them.
% Amongst a multitude of research directions, two themes most connected to this paper are: 
% \begin{enumeratea}
% 	\item {\bf Dynamics of networks:} Here one conceives of the network itself changing over time. Real world systems include social networks where new friendship ties are continuously created and some old ties dissolved; or information networks such as the Internet at the webpage or citation networks which again evolve over time. In this setting one aim of modeling such systems is to understand how simple probabilistic rules give rise to complex macroscopic structure, for example heavy tailed degree distribution.  
% 	\item {\bf Dynamics on networks:} Here one is interested in various dynamical processes on the network; prototypical examples include epidemic models such as the contact process or SIS epidemics, voters models, random walks and so on. Application areas range from understanding conditions on network structure and the parameters governing the rate of spread for a disease/opinion etc.\ to lead to an epidemic (infect a positive fraction of the network), the rate at which information diffuses through a network to understanding the performance of web-crawling algorithms, distributed computing on adaptive networks and so on. 
% \end{enumeratea} 
% It is hard to give a comprehensive set of references to this rapidly growing field, even to the sub-field of interest for this paper.  For rigorous results see e.g \cite{durrett-book,bollobas-book,janson-luczak-book,van2009random} and the references therein. To understand the range of questions being currently considered and their applications in a diverse array of fields, see \cite{newman2010networks,newman2003structure,albert2002statistical,barrat2008dynamical} and the references therein. In this context let us now describe the specific model of interest in this paper and then describe the motivations for such systems. 
% 
% 
% 
% In the context of applications, three prototypical examples include:
% 
% \begin{example}[{\bf Gossip algorithms \cite{bertsekas1989parallel,shah2009gossip}}]
% 	\label{eg:gossip}
% 	Quoting from the nice survey \cite{shah2009gossip},
% 	\emph{
% 	\begin{quote}
% 		`Unlike the Telephone network or the Internet, many of the next generation networks are not engineered for the purpose of providing efficient communication between various networked entities.  Indeed, these emerging networks do require algorithms for communication, computation, or merely spreading information. For example, estimation algorithms in sensor networks, broadcasting news through a peer-to-peer network, or viral advertising in a social network. These networks lack infrastructure; they exhibit unpredictable dynamics and they face stringent resource constraints. Therefore, algorithms operating within them need to be extremely simple, distributed, robust against networks dynamics, and efficient in resource utilization.'
% 	\end{quote}}
% 	In this context, two important examples are the class of \emph{linear computation} and \emph{separable function computation}. 
% 	In the first class, at time $t=0$ each vertex in the network starts with a real value $\set{x_i:i\in \Nbd}$.  
% 	The aim is to compute the average  $x_{\text{ave}} \doteq  n^{-1}\sum_{i\in \Nbd} x_i$. 
% 	However individuals can only communicate with each other through links in the network. 
% 	The second class of examples is morally similar: here each vertex $i\in \Nbd$ has a function $f_i$ and a value $x_i$. 
% 	The aim is to compute $\sum_{i\in \Nbd} f_i(x_i)$. 
% 	
% 	We now describe in more detail the context of linear computation although much of the discussion carries over to the second setting as well. For simplicity let us start with the case of fixed (non time-varying) network. Then set $X_0^{i,N} = x_i$ and 
% 	\[\bbar_{\alpha \gamma}(x,y) = x-y, \qquad \Rightarrow\; \;  b_{\alpha\gamma}(Z_s^{i, N}, \mu_s^{i,\gamma, N}) = \frac{\sum_{v\in \Nmc_i}(Z_s^{i,N} - Z_s^{v,N} )}{N_{i}}.\]
% %	\footnote{Ruoyu: Not sure which one to put here}
% %	\[\bbar_{\alpha \gamma}(x,y) = x-y, \qquad \Rightarrow\; \;  b_{\alpha\gamma}(Z_s^{i, N}, \mu_s^{i,\gamma, N}) = \frac{\sum_{j\in \Ngamma} \xi_{ij}^N(s) (Z_s^{i,N} - Z_s^{j,N} )}{N_{i,\gamma}(s)} \one_{\{N_{i,\gamma}(s)\}}.\]  
% \end{example}
% Thus in this case the values of neighboring vertices try to get `closer' to each other. Assuming the network is connected it is easy to show (see \cite{shah2009gossip}) that $(Z_t^{i, N}: i\in \Nbd) \to x_{\text{ave}}\boldsymbol{1}$ as $t\to\infty$. In the context of sensor networks, noise enters at many different stages. There is noise in transmission of data between vertices \cite{srivastava2010distributed,srivastava2011distributed} while the connectivity structure of the network itself could be changing over time \cite{nedic}. All of these motivate the above model. 
% 
% \begin{example}[{\bf Swarming}]
% 	\label{eg:swarming}
% 	A closely related model originates in the field of flocking or swarming dynamics to model the interaction of a collection of individuals such as schools of fish, human crowds or robots trying to co-ordinate movement. One standard model is through an aggregation model which incorporates a repulsive attractive force. Precisely suppose 
% 	\[\bbar_{\alpha\gamma}(x,y):= F_{\alpha\gamma}(|x - y|)\frac{x-y}{|x-y|}\]
% where $F_{\alpha\gamma}(\cdot):\mathbb{R}_+\to \mathbb{R} $ is an attractive-repulsive force, namely positive for small $r$ (so if particles get too close they are repelled) and negative for large $r$ (so particles typically tend to exhibit an attractive force unless they get too close). One example fitting our framework is 
% \[F_{\alpha\gamma}(r)  = \min(\kappa_{\alpha\gamma}r , 1-r)\] 
%    See \cite{bertozzi1} and the references therein for background on this model. 
% \end{example}
% 
% \begin{example}[{\bf Neuroscience}]
% 	\label{eg:neuro}
% 	
% 	Similar models arise frequently in neuroscience to study networks of spiking neurons (see \cite{BaladronFaugeras2012} and references therein).
% 	For example in \cite{BaladronFaugeras2012}, membrane potentials of neurons in the network are described by interacting diffusions and several variants of systems are proposed.
% 	The interaction in one of such systems corresponds to the functioning of chemical synapses, which is based on the transmission of certain neurotransmitter from presynaptic neuron to postsynaptic cell.
% 	The synaptic current induced by such chemical synapse is modeled as
% 	\begin{equation*}
% 		\bbar_{\alpha\gamma}(x,y) \doteq - \Jbar_{\alpha\gamma} (x_1 - V^{\alpha\gamma}_{\mbox{rev}}) y_3, \quad x=(x_1,x_2,x_3), y=(y_1,y_2,y_3) \in \Rmb^3.
% 	\end{equation*}
% 	Here $x_1 - V^{\alpha\gamma}_{\mbox{rev}}$ represents the voltage difference and $y_3$ denotes the fraction of certain open channels of the presynaptic neuron.
% 	For more details see \cite{BaladronFaugeras2012,Bossy2015clarification}.
% \end{example}
% 
% % We assume that the edge processes $\{\xi_{ij}^N\}$ are independent of the noise processes $\{W_i: i \in \Nmb\}$.
%  % and $\{\xi_{ij}^N(\cdot) :i\neq j\in [N]  \}$ are ${N \choose 2}$ `edge processes' represented by independent collections of $\{0,1\}$-valued i.i.d.\ stochastic processes describing whether {\bf direct} interaction occurs between $i$-th and $j$-th particles. 
% %The form of the coefficients and $\{\xi_{ij}^N\}$ in the SDE is such that the stochastic process $(Z^{1,N},\dotsc,Z^{N,N})$ is exchangeable if the initial distribution of the $N$ particles is exchangeable.
% 
% \subsection{Informal overview of our contributions}
% Since this paper straddles two different communities, the first being probabilistic combinatorics and the random graphs community and the second dealing with stochastic control and stochastic analysis, for the sake of understandability we deal with models which are both mathematically tractable and for which the proofs are amenable to both communities. In this section we give a brief overview both of the results in this paper as well as the new proof techniques developed to handle the model of interest.  Recall that the main motivation of this paper was to understand weakly interacting particle system models when one deviates from the complete graph. In this paper we focus on the study of Laws of Large numbers (LLN), Propagation of Chaos (POC) and Central Limit Theorems (CLT).
% %\textbf{This is motivated by...}


% Let us now say a few words about the proof techniques. The proof for the law of large numbers result follows by combining coupling arguments with a careful use of exchangeability of particles of each type and concentration inequalities for various functionals of the random graph.





%---------------------------------------------------------------------------------
%
%
% \section{Weak laws of large numbers}
%  \label{sec:model}
%
%
% Consider an infinite collection of particles where each particle belongs to one of $K$ different populations.
% Letting $\Kbd \doteq \{1,\dotsc,K\}$, define a function $\pbd : \Nmb \to \Kbd$ by $\pbd(i) \doteq \alpha$ if the $i$-th particle belongs to $\alpha$-th type or population.
% For $N \in \Nmb$, let $\Nbd = \{1,\dotsc,N\}$.
% For $\alpha \in \Kbd$, let $\Nalpha \doteq \{i \in \Nbd: \pbd(i) = \alpha \}$ be the number of particles of type $\alpha$ and denote by $N_\alpha$ the number of particles belonging to the $\alpha$-th population, namely $N_\alpha \doteq |\Nalpha|$.
% %Assume that $N_\alpha/N \to \Lambda_\alpha \in (0,1)$ as $N \to \infty$.
%
% %Assume that $\Nbar/N \to \bar\lambda \in (0,1)$ as $N \to \infty$.
%
%
% For fixed $N \in \Nmb$, consider the following system of equations for the $\Rd$-valued interacting diffusions $\{Z^{1,N},\dotsc,Z^{N,N}\}$, given on a filtered probability space $(\Omega,\Fmc,\Pbd,\{\Fmc_t\})$.
% %One can view above system of equations as a collection of classical weakly interacting particles where the interaction graph is not complete but a dynamical random graph with edges evolving independently.
% %This family of system is quite general in that the choice of random interaction graph includes any fixed \Erdos graph.
% %It also allows edges in the interaction graph to form and break up independently after an exponential time.

\section{Laws of large numbers}
\label{sec:laws-large-results}

%Conditions on the various coefficients will be introduced shortly.
We now describe our main results. This section deals with the law of large numbers while the next section concerns central limit theorems. 
Recall the collection of  SDE $\{Z_t^{i, N}: i\in \Nbd\}$ describing the evolution of $N$ interacting particles, defined in Section \ref{sec:mod} via \eqref{eq:ZiNt}. Along with this system, we will also consider a related infinite system of equations for $\Rd$-valued nonlinear diffusions $X^i$, $i \in \Nmb$, given on $(\Omega,\Fmc,\Pbd,\{\Fmc_t\})$.
Let $\Nmb_\alpha \doteq \{i \in \Nmb: \pbd(i) = \alpha \}$.
For $\alpha \in \Kbd$ and $i \in \Nmb_\alpha$,
\begin{equation} \label{eq:Xit}
	X^i_t = X^i_0 + \sum_{\gamma=1}^K \int_0^t b_{\alpha\gamma}(X^i_s,\mu^\gamma_s) \, ds + \sum_{\gamma=1}^K \int_0^t \sigma_{\alpha\gamma}(X^i_s,\mu^\gamma_s) \, dW^i_s, \quad \mu^\gamma_t = \Lmc(X^j_t), \quad i \in \Nmb_\alpha, j \in \Nmb_\gamma.
\end{equation}
%We assume that $\{X^i_0 : i \in \Nmb\}$ are i.i.d.\ $\Fmc_0$-measurable $\Rd$-valued random variables with distribution $\mu_0$.
The existence and uniqueness of pathwise solutions of \eqref{eq:ZiNt} and \eqref{eq:Xit} can be shown under the following conditions on the coefficients (cf.\ \cite{Sznitman1991}).

\begin{condition} \label{cond:coefficients}
	There exists some $L \in (0,\infty)$ such that for all $\alpha,\gamma \in \Kbd$, $\|\bbar_{\alpha\gamma}\|_{BL} \le L$ and $\|\sigmabar_{\alpha\gamma}\|_{BL} \le L$.
%	and $x,y,x',y' \in \Rmb^d$,
%	\begin{equation*}
%		\|\bbar_{\alpha\gamma}(x,y)\| \le L, \quad \|\beta(x,y) - \beta(x',y')\| \le L(\|x-x'\|+\|y-y'\|).
%	\end{equation*}
\end{condition}

We start with the following moment estimate. 
The proof is given in Section~\ref{sec:pf_NpN_rate}. For later use define $\Nbar \doteq \min_{\alpha \in \Kbd} N_\alpha$. 


\begin{theorem} \label{thm:NpN_rate}
	Suppose Condition~\ref{cond:coefficients} holds.
	Then
	\begin{equation} \label{eq:NpN rate}
		\sup_{N \ge 1} \max_{i \in \Nbd} \sqrt{\Nbar \pbar_N} \Ebd \|Z^{i,N} - X^i\|_{*,T}^2 < \infty.
	\end{equation}
\end{theorem}

\begin{remark}
	From the above theorem it is clear that $\Ebd \|Z^{i,N} - X^i\|_{*,T}^2$ is of order at most $(\Nbar \pbar_N)^{-1/2}$.
	If one assumes diffusion coefficients $\{\sigma_{\alpha\gamma}\}_{\alpha,\gamma \in \Kbd}$ to be constants, then the following result with a better order can be obtained:
	\begin{equation}
		\label{eq:NpN_better}
		\sup_{N \ge 1} \max_{i \in \Nbd} \sqrt{\Nbar \pbar_N} \Ebd \|Z^{i,N} - X^i\|_{*,T} < \infty.
	\end{equation}	
	See Remark \ref{rem:remsec6} for comments on this point.
\end{remark}

We will make the following assumption on $\pbar_N$. 
\begin{condition} \label{cond:cond1}
	$\Nbar \pbar_N \to \infty$ as $N \to \infty$.
\end{condition}

Theorem~\ref{thm:NpN_rate} together with a standard argument (cf.\ \cite{Sznitman1991}) implies that, under Conditions~\ref{cond:coefficients} and~\ref{cond:cond1}, the following propagation of chaos result holds.
We omit the proof.

\begin{corollary} \label{cor:poc}
	Suppose Conditions~\ref{cond:coefficients} and~\ref{cond:cond1} hold.
	Then for any $n$-tuple  $(i_1^N,\dotsc,i_n^N) \in \Nbd^n$ with $i_j^N \neq i_k^N$ whenever $j\neq k$, and
	$\pbd(i_j^N) = \alpha_j$, $j = 1,\dotsc,n$, 
	\begin{equation*}
		\Lmc(\{Z^{i_1^N,N},\dotsc,Z^{i_n^N,N}\}) \to \mu^{\alpha_1} \otimes \dotsb \otimes \mu^{\alpha_n}
	\end{equation*}
	in $\Pmc(\Cmc_d^n)$ as $N \to \infty$, where $\mu^\alpha \doteq \Lmc(X^i) \in \Pmc(\Cmc_d)$ for $\alpha \in \Kbd$ and $i \in \Nmb_\alpha$.
\end{corollary}



Using above results and an argument similar to~\cite{Sznitman1991} one can further show the following law of large numbers result. 
Proof is included in Appendix~\ref{sec:pf_Cor_LLN} for completeness.

\begin{corollary} \label{cor:LLN}
	Suppose Conditions~\ref{cond:coefficients} and~\ref{cond:cond1} hold.
	Then \\
	(a) For each $\alpha \in \Kbd$, as $N \to \infty$, $$\mu^{\alpha,N} \doteq \frac{1}{N_\alpha} \sum_{i \in \Nalpha} \delta_{Z^{i,N}} \Rightarrow \mu^\alpha.$$
	(b) Suppose in addition that we are in the setting of Example~\ref{eg:eg1}, namely $\xi_{ij}^N(t) \equiv \xi_{ij}^N(0)$ for all $t \ge 0$ and $i,j \in \Nbd$.
	Then for each $\alpha, \gamma \in \Kbd$ and $i \in \Nmb_\alpha$, as $N \to \infty$, $$\mu^{i,\gamma,N} \doteq \frac{1}{N_{i,\gamma}(0)} \sum_{j \in \Ngamma} \xi_{ij}^N(0) \delta_{Z^{j,N}} \one_{\{N_{i,\gamma}(0) > 0\}} \Rightarrow \mu^\gamma.$$
\end{corollary}




%----------------------------------------------------------




\section{Fluctuations and central limit theorems} \label{sec:fluctuations}

Next we will study the fluctuations of empirical measures about the law of large numbers limit.
For simplicity, we consider the {\em single-type} setting, i.e.\ $K=1$, and assume constant diffusion coefficients, i.e.\ $\sigma_{\alpha\gamma} \equiv I_d$, the $d$-dimensional identity matrix.
Consequently, we will write $\mu^N$, $\mu^{i,N}$, $\mu$, $N_i(t)$, $p_N(t)$, $b$ and $\bbar$ instead of $\mu^{\alpha,N}$, $\mu^{i,\gamma,N}$, $\mu^\alpha$, $N_{i,\gamma}(t)$, $p_{\alpha\gamma,N}(t)$, $b_{\alpha\gamma}$ and $\bbar_{\alpha\gamma}$.
Also, to simplify the notation, we will abbreviate $\xi_{ij}^N(t)$ and $\xi_{ij}^N$ as $\xi_{ij}(t)$ and $\xi_{ij}$ in the rest of the paper.

% and assume the setting of Remark \ref{rmk:edge_setting}$(a)$ throughout.

\subsection{Canonical processes} \label{sec:canonical_processes}

We first introduce the following canonical spaces and stochastic processes.
Let $\Omega_d \doteq \Cmc_d \times \Cmc_d$, $\Omega_e \doteq \Dmb([0,T]:\{0,1\})$ and $\Omega_N \doteq \Omega_d^N \times \Omega_e^{N \times N}$. 
Denote by $\nu \in \Pmc(\Omega_d)$ the common law of $(W^i, X^i)$ where $i \in \Nbd$ and $X^i$ is given by \eqref{eq:Xit}
(under the setting of this section).
Also denote by $\nu_{e,N} \in [\Pmc(\Omega_e)]^{N \times N}$ the law of the random adjacency matrix process $\{\xi_{ij}(t) : i,j \in \Nbd, t \in [0,T] \}$.
% with $\xi_{ii}(t) \equiv 1$ for $i \in \Nbd$ and $\xi_{ij} = \xi_{ji}$ being i.i.d.\ Bernoulli$(N,p_N)$ for $1 \le i < j \le N$.   
Define for $N \in \Nmb$ the probability measure $\Pmb^N$ on $\Omega_N$ as
\begin{equation*}
	\Pmb^{N} \doteq \Lmc \left((W^1, X^1), (W^2, X^2), \dotsc,(W^N, X^N), \{\xi_{ij} : i,j \in \Nbd \} \right) \equiv \nu^{\otimes N} \otimes \nu_{e,N}.
\end{equation*}
%namely $\Pmb^N = \nu^{\otimes N} \otimes \nu_{e,N}$.
%Expected value under $\Pmb^N$ will be denoted as $\Ebf_{\Pmb^N}$.
For $\omega = (\omega_1, \omega_2, \dotsc, \omega_N, \omegabar) \in \Omega_N$ with $\omegabar = (\omegabar_{ij})_{1 \le i,j \le N}$, let $V^i(\omega) \doteq \omega_i, i \in \Nbd$ and abusing notation,
\begin{equation*}
	V^i \doteq (W^i, X^i), \quad \xi_{ij}(\omega) \doteq \omegabar_{ij}, \quad i,j \in \Nbd.
\end{equation*}
Also define the canonical processes $V_* \doteq (W_*, X_*)$ on $\Omega_d$ as
\begin{equation*}
	V_*(\omega) \doteq (W_*(\omega), X_*(\omega)) \doteq (\omega_1, \omega_2), \quad \omega = (\omega_1, \omega_2) \in \Omega_d.
\end{equation*}




%----------------------------------------------------------



\subsection{Some integral operators} \label{sec:some_integral_operators}

We will need the following functions for stating our central limit theorem. Recall that $\mu = \nu_{(2)}$ denotes the law of $X^i$. Let $\mu_t$ be the marginal of $\mu$ at instant $t$, namely, $\mu_t \doteq \Lmc(X^i_t)$.
Define for $t \in [0,T]$, function $\bbar_t$ from $\Rmb^d \times \Rmb^d$ to $\Rmb^d$ as
\begin{equation} \label{eq:bbar t}
	\bbar_t(x,y) \doteq \bbar(x,y) - \int_{\Rmb^d} \bbar(x,z)\mu_t(dz), \quad (x,y) \in \Rmb^d \times \Rmb^d.
\end{equation}
Define function $h$ from $\Omega_d \times \Omega_d$ to $\Rmb$ ($\nu \otimes \nu$ a.s.) as
\begin{equation} \label{eq:h}
	h(\omega,\omega') \doteq \int_0^T \bbar_t (X_{*,t}(\omega),X_{*,t}(\omega')) \cdot dW_{*,t}(\omega), \quad (\omega, \omega') \in \Omega_d \times \Omega_d.
\end{equation}
Now consider the Hilbert space $L^2(\Omega_d, \nu)$.
Define integral operators $A$ on $L^2(\Omega_d, \nu)$ as
\begin{equation} \label{eq:A} 
	A f(\omega) \doteq \int_{\Omega_d} h(\omega',\omega) f(\omega') \, \nu(d\omega'), \quad f \in L^2(\Omega_d, \nu), \: \omega \in \Omega_d.
\end{equation}
Denote by $I$ the identity operator on $L^2(\Omega_d, \nu)$.
For $t \in [0,T]$, let
\begin{equation} \label{eq:lambdas}
	\lambda_t \doteq \int_{\Rd \times \Rd} \| \bbar_t(x,y) \|^2 \, \mu_t(dx) \, \mu_t(dy).
\end{equation} 
The following lemma is taken from \cite{BudhirajaWu2015} (see Lemma $3.1$ therein).
%The following lemma will be proved in Section \ref{sec:Proof of Lemma of Trace}.

\begin{lemma} \label{lem:trace}
	(a) $\textnormal{Trace}(AA^*) = \int_{\Omega_d^2} h^2(\omega,\omega') \, \nu(d\omega) \, \nu(d\omega') = \int_0^T \lambda_t \, dt$.
	(b) \textnormal{Trace}$(A^n) = 0$ for all $n \ge 2$.
	(c) $I - A$ is invertible.
\end{lemma}



%----------------------------------------------------------



\subsection{Central limit theorem} \label{sec:CLT}

For the central limit theorem we need the following strengthened version of Condition \ref{cond:cond1}.

\begin{condition} \label{cond:cond2}
	$K=1$, $\sigma_{\alpha\gamma} \equiv I_d$, where $I_d$ is the $d$-dimensional identity matrix. The collection
	$\{ p_N(\cdot)  \}_{N \in \Nmb}$ is pre-compact in $\Cmb([0,T]:\Rmb)$ and $\liminf_{N \to \infty} \pbar_N > 0$.	
	% \begin{equation*}
	% 		\lim_{N \to \infty} \|p_N(\cdot) - p(\cdot)\|_{*,T} = 0, \mbox{ and } \pbar \doteq \inf_{t \in [0,T]} p(t) > 0. 
%\end{equation*}
\end{condition}
We can now present the central limit theorem.
Let $L^2_c(\Cmc_d,\mu)$ be the space of all functions $\phi \in L^2(\Cmc_d,\mu)$ such that $\lan \phi, \mu \ran = 0$.
For $\phi \in L^2_c(\Cmc_d,\mu)$, let $\eta^N (\phi) \doteq \frac{1}{\sqrt{N}} \sum_{i=1}^N \phi (Z^{i,N})$ and $\phibd \doteq \phi(X_*) \in L^2(\Omega_d, \nu)$.
%Denote by $\Amc$ the collection of all measurable maps $\phi \colon \Cmc_d \to \Rmb$ such that $\phi(X_*) \in L^2(\Omega_d, \nu)$ and $\int_{\Omega_d} \phi(X_*(\omega)) \nu(d\omega) = 0$, namely $\lan \phi^2,\mu \ran < \infty$ and $\lan \phi,\mu \ran = 0$.
%For $\phi \in \Amc$, let 

\begin{theorem} \label{thm:CLT}
	Suppose Conditions \ref{cond:coefficients} and \ref{cond:cond2} hold.
	Then $\{\eta^N (\phi) : \phi \in L^2_c(\Cmc_d,\mu)\}$ converges as $N \to \infty$ to a mean $0$ Gaussian field $\{\eta(\phi) : \phi \in L^2_c(\Cmc_d,\mu)\}$ in the sense of convergence of finite dimensional distributions, where for $\phi,\psi \in L^2_c(\Cmc_d,\mu)$,
	\begin{equation*}
		\Ebf [\eta(\phi) \eta(\psi)] = \langle (I-A)^{-1} \phibd, (I-A)^{-1} \psibd \rangle_{L^2(\Omega_d, \nu)}.
	\end{equation*}
%	and , $\psibd \doteq \psi(X_*) \in L^2(\Omega_d, \nu)$.
\end{theorem}

Proof of the theorem is given in Section \ref{sec:pf_CLT}.



%----------------------------------------------------------


% \section{Discussion and related literature}
% \label{sec:disc}
% 
% As described in the examples considered in Section \ref{sec:mod}, one can view $\{\xi_{ij}^N\}$ as a collection of edges making up a dynamic random interaction graph, which influence the trajectories of $N$ individuals (people, animals, computers, neurons etc.) $\{Z^{1,N},\dotsc,Z^{N,N}\}$ through the prescribed connectivity structure.
% When this interaction graph is always complete, i.e.\ $\xi_{ij}^N(t) \equiv 1$ for all $i,j$, the empirical distribution of $i$-th particle's $\gamma$-th type neighbors $\mu^{i,\gamma,N}_t$ is the same as that of all $\gamma$-th type particles $\mu^{\gamma,N}_t \doteq \frac{1}{N_\gamma} \sum_{j\in\Ngamma} \delta_{Z^{j,N}_t}$, where recall $N_\gamma$ denotes the number of $\gamma$-th type particles for $\gamma \in \Kbd$. 
% Such systems  are commonly referred to as \textit{weakly interacting diffusion processes}, and date back to classical works of Boltzmann, Vlasov, McKean and others (see \cite{Sznitman1991,Kolokoltsov2010} and references therein) and have long been of interest in statistical physics.
% In recent years, such models have arisen in many different application areas, ranging from mathematical finance and chemical and biological systems to communication networks and social sciences (see \cite{BudhirajaDupuisFischerRamanan2015limits} and a list of references therein).
% In the complete graph setting, a host of results have been proven including laws of large numbers (LLN), propagation of chaos (POC) properties, and central limit theorems (CLT), see e.g.\ McKean~\cite{McKean1966class,McKean1967propagation}, Braun and Hepp~\cite{BraunHepp1977vlasov}, Dawson~\cite{Dawson1983critical}, Tanaka~\cite{Tanaka1984limit}, Oelschal\"{a}ger~\cite{Oelschlager1984martingale}, Sznitman~\cite{Sznitman1984,Sznitman1991}, Graham and M\'{e}l\'{e}ard~\cite{GrahamMeleard1997}, Shiga and Tanaka~\cite{ShigaTanaka1985}, M\'{e}l\'{e}ard~\cite{Meleard1998}.
% However all of these  papers consider exchangeable populations, i.e.\ $K=1$, and complete interaction graphs where every vertex influences every other vertex (albeit in a `weak' fashion). A number of variations have also been proposed.
% For example, in~\cite{KurtzXiong1999,KurtzXiong2004,BudhirajaSaha2014,BudhirajaWu2015} a common noise process that influences the dynamics of each particle is introduced to the system and limit theorems of the above form are established.
% For a setting with $K$-different subpopulations within each of which particle evolution is exchangeable, LLN and POC have been studied in~\cite{BaladronFaugeras2012}, and a corresponding CLT has been established in~\cite{BudhirajaWu2015}.
% Other works that have studied mean field results for heterogeneous populations include~\cite{Collet2014macroscopic,ChongKluppelberg2015partial}.
%  
% The main aim of this work was to understand the following two questions:
% \begin{enumeratea}
% 	\item {\bf Random graph setting:} What if one does not have a complete interaction graph. How far do classical results carry over, especially in the context of inhomogeneous random graph models such as the fundamental  Bollobas-Janson-Riordan random graph model \cite{bollobas-riordan-janson}?  More importantly what new mathematical techniques are required to prove such results?
% 	\item {\bf Dynamic random graphs:} what if the interaction graph is dynamic and evolves over time? How far do classical results carry over in this context?
% \end{enumeratea}


\section{Preliminary estimates} \label{sec:pre_results}

In this section we present several elementary results for a binomial distribution, which will be used for the proof of Theorems~\ref{thm:NpN_rate} and~\ref{thm:CLT}.
Proofs to these results are provided in Appendix~\ref{sec:pf_section_5} for completeness.
%Denote by $Bin(n,p)$ the Binomial distribution with number of trials $n$ and probability of success $p$.

\begin{lemma} 
	\label{lem:prep_1}
	Let $X$ be a Binomial random variable with number of trials $n$ and probability of success $p$.
	Let $q \doteq 1 - p$.
	Then
	\begin{equation*}
		\Ebf \frac{1}{X+1} = \frac{1-q^{n+1}}{(n+1)p} \le \frac{1}{(n+1)p}.
	\end{equation*}
	Also for each $m = 2,3,\dotsc$,
	\begin{align*}
		& \Ebf \frac{1}{X+m} \le \frac{1-q^{n+1}}{(n+m)p} \le \frac{1}{(n+m)p}, \\
		& \Ebf \frac{1}{(X+1)^m} \le \frac{m^m}{(n+1)^m p^m}.
	\end{align*}
\end{lemma}

For the following lemma, let $\zeta_{ii} \equiv 1$ for $i \in \Nbd$ and $\{\zeta_{ij} = \zeta_{ji} : 1 \le i < j \le N\}$ be independent Bernoulli random variables with $\Pbd(\zeta_{ij}=1) = p_{\alpha\gamma,N} = p_{\gamma\alpha,N}$ for $i \in \Nalpha$, $j \in \Ngamma$, and $\alpha, \gamma \in \Kbd$.
Let $q_{\alpha\gamma,N} \doteq 1 - p_{\alpha\gamma,N}$ and $N_{i,\gamma} \doteq \sum_{j \in \Ngamma} \zeta_{ij}$ for $i \in \Nbd$ and $\alpha, \gamma \in \Kbd$. 

%\begin{definition} \label{def:def1}
%	For $i \in \Nmb$, let $\Ntil_{i,-i} = \sum_{j \ne i}^N \zeta_{ij} = N_i - 1$. 
%	For $i,k \in \Nmb$ and $i \ne k$, let $\Ntil_{i,-ik} = \sum_{j \notin \{i,k\}} \zeta_{ij}$. 
%	For $i,j,k \in \Nmb$ distinct, let $\Ntil_{i,-ijk} = \sum_{l \notin \{i,j,k\}} \zeta_{il}$.
%\end{definition}

\begin{lemma} 
	\label{lem:prep_2}
	For  $\alpha,\gamma \in \Kbd$,
	\begin{align*}
		& \Ebf \left( \sum_{k \in \Nalpha} \frac{N_\gamma \zeta_{ki_\gamma}}{N_\alpha N_{k,\gamma}} \one_{\{N_{k,\gamma} > 0\}} - 1 \right)^2 \le \frac{4}{N_\alpha p_{\alpha\gamma,N}} + 2 e^{-N_\gamma p_{\alpha\gamma,N}}, \quad i_\gamma \in \Ngamma, \\
		& \Ebf \left( \sum_{k \in \Nalpha} \frac{\zeta_{ki_\alpha}}{N_{k,\alpha}} - 1 \right)^2 \le \frac{3}{N_\alpha p_{\alpha\alpha,N}}, \quad i_\alpha \in \Nalpha.
	\end{align*}
\end{lemma}


%We also have the following expression and bounds for on certain expectations.
%
%\begin{lemma} 
%	\label{lem:prep_3}
%	For different $i,j \in \Nbd$,
%	\begin{enumerate}[(a)]
%	\item
%		$\displaystyle \Ebd \frac{1}{N_i} = \frac{1}{N} + \Rmc_N$,
%	\item
%		$\displaystyle \Ebd \frac{\zeta_{ij}}{N_i} = \frac{1}{N} - \frac{\Rmc_N}{N-1}$,
%	\item
%		$\displaystyle \sum_{j=1}^N \left(\Ebd \frac{\zeta_{ij}}{N_i}\right)^2 = \frac{1}{N} + \frac{N}{N-1} \Rmc_N^2 \le \frac{1}{N p_N}$,
%	\end{enumerate}
%	where $\Rmc_N \doteq \Rmc(p_N) \doteq \frac{q_N(1-q_N^{N-1})}{N p_N} \le \frac{1}{N p_N}$.
%\end{lemma}


For the following lemma, let $Y = 1 + \sum_{i=2}^N \zeta_i$, where $\{\zeta_i, i=2,\dotsc,N\}$ are i.i.d.\ Bernoulli random variables with probability of success $p_N$.
We have the following tail bound on the random variable $Y$.
\begin{lemma} 
	\label{lem:prep_4}
	For $k > 0$, let $C_N(k) \doteq \sqrt{k (N-1) \log N }$. Then
	$\Pbd \left( |Y - N p_N| > C_N(k) + 1 \right) \le \frac{2}{N^{2k}}.$
\end{lemma}
\section{Proof of Theorem \ref{thm:NpN_rate}} \label{sec:pf_NpN_rate}
Fix $\alpha \in \Kbd$ and $\ialpha \in \Nalpha$.
%	For simplicity, assume without loss of generality that $\ialpha = \alpha$ for $\alpha \in \Kbd$.
	For fixed $t \in [0,T]$, we have from Cauchy--Schwarz and Doob's inequalities that
	\begin{align}
		\Ebf \left\|Z^{\ialpha,N} - X^{\ialpha}\right\|_{*,t}^2 
		& \le 2KT \sum_{\gamma=1}^K \int_0^t \Ebf \left\| b_{\alpha\gamma}(Z^{\ialpha,N}_s, \mu^{\ialpha,\gamma,N}_s) - b_{\alpha\gamma}(X^{\ialpha}_s, \mu^\gamma_s) \right\|^2 ds \notag \\
		& \qquad + 8K \sum_{\gamma=1}^K \int_0^t \Ebf \left\| \sigma_{\alpha\gamma}(Z^{\ialpha,N}_s, \mu^{\ialpha,\gamma,N}_s) - \sigma_{\alpha\gamma}(X^{\ialpha}_s, \mu^\gamma_s) \right\|^2 ds. \label{eq:NpN_1}
%		& \le 2KT \sum_{\gamma=1}^K \int_0^t \Ebf \left\| \sum_{j \in \Ngamma} \frac{\xi_{\ialphaj}(s)}{N_{1,\gamma}(s)} \one_{\{N_{1,\gamma}(s) > 0\}} \bbar_{\alpha\gamma}(Z^{\ialpha,N}_s,Z^{j,N}_s) - b_{\alpha\gamma}(X^{\ialpha}_s,\mu^\gamma_s) \right\|^2 ds \notag \\
%		& \qquad + 8KT \sum_{\gamma=1}^K \int_0^t \Ebf \left\| \sum_{j \in \Ngamma} \frac{\xi_{\ialphaj}(s)}{N_{1,\gamma}(s)} \one_{\{N_{1,\gamma}(s) > 0\}} \sigmabar_{\alpha\gamma}(Z^{\ialpha,N}_s,Z^{j,N}_s) - \sigma_{\alpha\gamma}(X^{\ialpha}_s,\mu^\gamma_s) \right\|^2 ds. \label{eq:NpN_1}
	\end{align}
	Fix $\gamma \in \Kbd$.
	Note that by adding and subtracting terms we have for $s \in [0,T]$,
	\begin{align}
		& \left\| b_{\alpha\gamma}(Z^{\ialpha,N}_s, \mu^{\ialpha,\gamma,N}_s) - b_{\alpha\gamma}(X^{\ialpha}_s, \mu^\gamma_s) \right\| \notag \\
		& \quad = \left\| \sum_{j \in \Ngamma} \frac{\xi_{\ialpha j}(s)}{N_{\ialpha,\gamma}(s)} \one_{\{N_{\ialpha,\gamma}(s) > 0\}} \bbar_{\alpha\gamma}(Z^{\ialpha,N}_s,Z^{j,N}_s) - b_{\alpha\gamma}(X^{\ialpha}_s,\mu^\gamma_s) \right\| \notag \\
		& \quad \le \sum_{n=1}^4 \Tmc^{\gamma,N,n}(s), \label{eq:NpN_2}
			\end{align}
where
$$		
\Tmc^{\gamma,N,1}(s) \doteq 		\left\| \sum_{j \in \Ngamma} \frac{\xi_{\ialpha j}(s)}{N_{\ialpha,\gamma}(s)} \one_{\{N_{\ialpha,\gamma}(s) > 0\}} \left( \bbar_{\alpha\gamma}(Z^{\ialpha,N}_s,\ZjNs) - \bbar_{\alpha\gamma}(X^{\ialpha}_s,\ZjNs) \right) \right\|,$$
$$
\Tmc^{\gamma,N,2}(s) \doteq \left\| \sum_{j \in \Ngamma} \frac{\xi_{\ialpha j}(s)}{N_{\ialpha,\gamma}(s)} \one_{\{N_{\ialpha,\gamma}(s) > 0\}} \left( \bbar_{\alpha\gamma}(X^{\ialpha}_s,\ZjNs) - \bbar_{\alpha\gamma}(X^{\ialpha}_s,X^j_s) \right) \right\|,$$
$$
\Tmc^{\gamma,N,3}(s) \doteq		\left\| \sum_{j \in \Ngamma} \frac{\xi_{\ialpha j}(s)}{N_{\ialpha,\gamma}(s)} \one_{\{N_{\ialpha,\gamma}(s) > 0\}} \left(\bbar_{\alpha\gamma}(X^{\ialpha}_s,X^j_s) - b_{\alpha\gamma}(X^{\ialpha}_s,\mu^\gamma_s) \right) \right\|,$$
and
$$
\Tmc^{\gamma,N,4}(s) \doteq \left\| \left( \sum_{j \in \Ngamma} \frac{\xi_{\ialpha j}(s)}{N_{\ialpha,\gamma}(s)} \one_{\{N_{\ialpha,\gamma}(s) > 0\}} - 1 \right) b_{\alpha\gamma}(X^{\ialpha}_s,\mu^\gamma_s) \right\|.$$
	
	Now we will analyze these terms one by one.
	For $\Tmc^{\gamma,N,1}$, first note that
	\begin{equation*}
		\sum_{j \in \Ngamma} \frac{\xi_{\ialpha j}(s)}{N_{\ialpha,\gamma}(s)} \one_{\{N_{\ialpha,\gamma}(s) > 0\}} = \one_{\{N_{\ialpha,\gamma}(s) > 0\}} \le 1.
	\end{equation*}
	So we have from Cauchy--Schwarz inequality and Lipschitz property of $\bbar$ that
	\begin{align}
		\Ebf [\Tmc^{\gamma,N,1}(s)]^2 & = \Ebf \left\| \sum_{j \in \Ngamma} \frac{\xi_{\ialpha j}(s)}{N_{\ialpha,\gamma}(s)} \one_{\{N_{\ialpha,\gamma}(s) > 0\}} \left( \bbar_{\alpha\gamma}(Z^{\ialpha,N}_s,\ZjNs) - \bbar_{\alpha\gamma}(X^{\ialpha}_s,\ZjNs) \right) \right\|^2 \notag \\
%		& \le \Ebf \sum_{j \in \Ngamma} \frac{\xi_{\ialpha j}(s)}{N_{\ialpha,\gamma}(s)} \one_{\{N_{\ialpha,\gamma}(s) > 0\}} \left\| \bbar(Z^{\ialpha,N}_s,Z^{j,N}_s) - \bbar(X^{\ialpha}_s,Z^{j,N}_s) \right\|^2 \notag %\\
		& \le L^2 \Ebf \sum_{j \in \Ngamma} \frac{\xi_{\ialpha j}(s)}{N_{\ialpha,\gamma}(s)} \one_{\{N_{\ialpha,\gamma}(s) > 0\}} \left\| Z^{\ialpha,N}_s - X^{\ialpha}_s \right\|^2 \notag \\
		& \le L^2 \Ebf \|Z^{\ialpha,N}_s - X^{\ialpha}_s\|^2. \label{eq:TN1_bd}
	\end{align}
	
	For $\Tmc^{\gamma,N,2}$, when $\gamma \ne \alpha$, we have
	\begin{align}
		\Ebf [\Tmc^{\gamma,N,2}(s)]^2 & = \Ebf \left\| \sum_{j \in \Ngamma} \frac{\xi_{\ialpha j}(s)}{N_{\ialpha,\gamma}(s)} \one_{\{N_{\ialpha,\gamma}(s) > 0\}} \left( \bbar_{\alpha\gamma}(X^{\ialpha}_s,\ZjNs) - \bbar_{\alpha\gamma}(X^{\ialpha}_s,X^j_s) \right) \right\|^2 \notag \\
%		& \le \Ebf \sum_{j \in \Ngamma} \frac{\xi_{\ialphaj}(s)}{N_{\ialpha,\gamma}(s)} \one_{\{N_{\ialpha,\gamma}(s) > 0\}} \left\| \bbar_{\alpha\gamma}(X^{\ialpha}_s,\ZjNs) - \bbar_{\alpha\gamma}(X^{\ialpha}_s,X^j_s) \right\| \notag \\
		& \le L^2 \Ebf \sum_{j \in \Ngamma} \frac{\xi_{\ialpha j}(s)}{N_{\ialpha,\gamma}(s)} \one_{\{N_{\ialpha,\gamma}(s) > 0\}} \left\|Z^{j,N}_s - X^j_s\right\|^2 \notag \\
		& = L^2 \Ebf \frac{N_\gamma \xi_{\ialpha\igamma}(s)}{N_{\ialpha,\gamma}(s)} \one_{\{N_{\ialpha,\gamma}(s) > 0\}} \left\|Z^{\igamma,N}_s - X^\igamma_s\right\|^2 \notag \\
		& = L^2 \Ebf \sum_{k \in \Nalpha} \frac{N_\gamma \xi_{k\igamma}(s)}{N_\alpha N_{k,\gamma}(s)} \one_{\{N_{k,\gamma}(s) > 0\}} \left\|Z^{\igamma,N}_s - X^\igamma_s\right\|^2, \label{eq:TN2_gamma}
	\end{align}
	where the last two equalities follow from the fact that when $\gamma \ne \alpha$
	\begin{align*}
		\Lmc(\xi_{i_\alpha j}(s), N_{i_\alpha,\gamma}(s), Z^{j,N}_s, X^j_s) & = \Lmc(\xi_{i_\alpha i_\gamma}(s), N_{i_\alpha,\gamma}(s), Z^{\igamma,N}_s, X^\igamma_s) \\
		& = \Lmc(\xi_{ki_\gamma}(s), N_{k,\gamma}(s), Z^{\igamma,N}_s, X^\igamma_s), \quad j \in \Ngamma, k \in \Nalpha.
	\end{align*}
	When $\gamma = \alpha$, we have
	\begin{align}
		\Ebf [\Tmc^{\alpha,N,2}(s)]^2 & = \Ebf \left\| \sum_{k \in \Nalpha} \frac{\xi_{\ialpha k}(s)}{N_{\ialpha,\alpha}(s)} \left( \bbar_{\alpha\alpha}(X^{\ialpha}_s,Z^{k,N}_s) - \bbar_{\alpha\alpha}(X^{\ialpha}_s,X^k_s) \right) \right\|^2 \notag \\
%		& \le \Ebf \sum_{k \in \Nalpha} \frac{\xi_{\ialphak}(s)}{N_{\ialpha,1}(s)} \one_{\{N_{\ialpha,1}(s) > 0\}} \left\| \bbar_{11}(X^{\ialpha}_s,\ZkNs) - \bbar_{11}(X^{\ialpha}_s,X^k_s) \right\| \notag \\
		& \le L^2 \Ebf \sum_{k \in \Nalpha} \frac{\xi_{\ialpha k}(s)}{N_{\ialpha,\alpha}(s)} \left\|Z^{k,N}_s - X^k_s\right\|^2 \notag \\
%		& = L^2 \Ebf \frac{N_\alpha \xi_{\ialpha1}(s)}{N_{\ialpha,1}(s)} \one_{\{N_{\ialpha,1}(s) > 0\}} \left\|Z^{\ialpha,N}_s - X^1_s\right\|^2 \notag \\
		& = L^2 \Ebf \sum_{k \in \Nalpha} \frac{\xi_{k\ialpha}(s)}{N_{k,\alpha}(s)} \left\|Z^{\ialpha,N}_s - X^\ialpha_s\right\|^2, \label{eq:TN2_alpha}
%		& = L^2 \Ebf \left[ \left( \sum_{k \in \Nalpha} \frac{\xi_{k\ialpha}(s)}{N_{k,\alpha}(s)} - 1 \right) \left\|Z^{\ialpha,N}_s - X^\ialpha_s\right\|^2 \right] + L^2 \Ebf \left\|Z^{\ialpha,N}_s - X^\ialpha_s\right\|^2 \notag \\
%		& \le L^2 \sqrt{ \Ebf \left( \sum_{k \in \Nalpha} \frac{\xi_{k\ialpha}(s)}{N_{k,\alpha}(s)} - 1 \right)^2 \Ebf \left\|Z^{\ialpha,N}_s - X^\ialpha_s\right\|^4} + L^2 \Ebf \left\|Z^{\ialpha,N}_s - X^\ialpha_s\right\|^2.
%		& \le 2 K^2 s \sqrt{\Ebf \left( \sum_{k=1}^N \frac{\xi_{\ialphak}(s)}{N_k(s)} - 1 \right)^2} + K \Ebf \left\|Z^{\ialpha,N}_s - X^{\ialpha}_s\right\|, 	
	\end{align}	
	where the last line follows from the fact that
	\begin{equation*}
		\Lmc(\xi_{\ialpha k}(s), N_{\ialpha,\alpha}(s), Z^{k,N}_s, X^k_s) = \Lmc(\xi_{k\ialpha}(s), N_{k,\alpha}(s), Z^{\ialpha,N}_s, X^{\ialpha}_s), \quad k \in \Nalpha.
	\end{equation*}
	Combining \eqref{eq:TN2_gamma} and \eqref{eq:TN2_alpha} shows that for each $\gamma \in \Kbd$,
	\begin{align}
		& \Ebf [\Tmc^{\gamma,N,2}(s)]^2 \notag \\
		& \le L^2 \Ebf \sum_{k \in \Nalpha} \frac{N_\gamma \xi_{k\igamma}(s)}{N_\alpha N_{k,\gamma}(s)} \one_{\{N_{k,\gamma}(s) > 0\}} \left\|Z^{\igamma,N}_s - X^\igamma_s\right\|^2 \notag \\
		& = L^2 \Ebf \left[ \left( \sum_{k \in \Nalpha} \frac{N_\gamma \xi_{k\igamma}(s)}{N_\alpha N_{k,\gamma}(s)} \one_{\{N_{k,\gamma}(s) > 0\}} - 1 \right) \left\|Z^{\igamma,N}_s - X^\igamma_s\right\|^2 \right] + L^2 \Ebf \left\|Z^{\igamma,N}_s - X^\igamma_s\right\|^2 \notag \\
		& \le L^2 \left[ \Ebf \left( \sum_{k \in \Nalpha} \frac{N_\gamma \xi_{k\igamma}(s)}{N_\alpha N_{k,\gamma}(s)} \one_{\{N_{k,\gamma}(s) > 0\}} - 1 \right)^2 \Ebf \left\|Z^{\igamma,N}_s - X^\igamma_s\right\|^4\right]^{1/2} + L^2 \Ebf \left\|Z^{\igamma,N}_s - X^\igamma_s\right\|^2. \label{eq:TN2_temp}
	\end{align}
	From Condition \ref{cond:coefficients} and an application of Cauchy--Schwarz and Doob's inequalities analogous to that for \eqref{eq:NpN_1} it follows that
	\begin{equation} 
		\label{eq:ZiXi_bd}
		\sup_{N \in \Nmb} \max_{i \in \Nbd} \Ebf \left\|Z^{i,N}_t - X^i_t\right\|_{*,T}^4 \doteq \kappa_1 < \infty.
	\end{equation}
	Applying this and Lemma \ref{lem:prep_2} to \eqref{eq:TN2_temp} gives us for all $\gamma \in \Kbd$
	\begin{equation} 
		\label{eq:TN2_bd}
		\Ebf [\Tmc^{\gamma,N,2}(s)]^2 \le \kappa_2 \left(\frac{1}{N_\alpha p_{\alpha\gamma,N}(s)} + e^{-N_\gamma p_{\alpha\gamma,N}(s)}\right)^{1/2} + L^2 \Ebf \left\|Z^{\igamma,N}_s - X^{\igamma}_s\right\|^2.
	\end{equation}
	
	For $\Tmc^{\gamma,N,3}$, since $\{X^i: i \in \Nbd\}$ are independent of $\{\xi_{ij}, N_{i,\gamma}: i,j \in \Nbd, \gamma \in \Kbd\}$, we have
	\begin{align*}
		\Ebf \left[\Tmc^{\gamma,N,3}(s)\right]^2 & = \Ebf \left\| \sum_{j \in \Ngamma} \frac{\xi_{\ialpha j}(s)}{N_{\ialpha,\gamma}(s)} \one_{\{N_{\ialpha,\gamma}(s) > 0\}} \left(\bbar_{\alpha\gamma}(X^{\ialpha}_s,X^j_s) - b_{\alpha\gamma}(X^{\ialpha}_s,\mu^\gamma_s) \right) \right\|^2  \\
		& = \sum_{j \in \Ngamma} \sum_{k \in \Ngamma} \left[ \Ebf \left( \frac{\xi_{\ialpha j}(s)}{N_{\ialpha,\gamma}(s)} \one_{\{N_{\ialpha,\gamma}(s) > 0\}} \frac{\xi_{\ialpha k}(s)}{N_{\ialpha,\gamma}(s)} \one_{\{N_{\ialpha,\gamma}(s) > 0\}} \right) \right. \\
		& \quad \cdot \left. \Ebf \left( \left(\bbar_{\alpha\gamma}(X^{\ialpha}_s,X^j_s) - b_{\alpha\gamma}(X^{\ialpha}_s,\mu^\gamma_s) \right) \left(\bbar_{\alpha\gamma}(X^{\ialpha}_s,X^k_s) - b_{\alpha\gamma}(X^{\ialpha}_s,\mu^\gamma_s) \right) \right) \right]  \\
		& = \sum_{j \in \Ngamma} \left[ \Ebf \left( \frac{\xi_{\ialpha j}(s)}{N_{\ialpha,\gamma}^2(s)} \one_{\{N_{\ialpha,\gamma}(s) > 0\}} \right) \Ebf \left( \bbar_{\alpha\gamma}(X^{\ialpha}_s,X^j_s) - b_{\alpha\gamma}(X^{\ialpha}_s,\mu^\gamma_s) \right)^2 \right].
	\end{align*}
	Thus using Lemma \ref{lem:prep_1},
	\begin{align}
		\Ebf \left[\Tmc^{\gamma,N,3}(s)\right]^2
		& \le 4L^2 \Ebf \sum_{j \in \Ngamma} \frac{\xi_{\ialpha j}(s)}{N_{\ialpha,\gamma}^2(s)} \one_{\{N_{\ialpha,\gamma}(s) > 0\}}  = 4L^2 \Ebf \frac{1}{N_{\ialpha,\gamma}(s)} \one_{\{N_{\ialpha,\gamma}(s) > 0\}} \notag \\
%		& \le K^2 \sum_{j=1}^N \Ebf \left(\frac{\xi_{\ialphaj}(s)}{N_\alpha(s)} - \Ebf \frac{\xi_{\ialphaj}(s)}{N_\alpha(s)}\right)^2 \\
		& \le 4L^2 \Ebf \frac{2}{1 + N_{\ialpha,\gamma}(s)}  \le \frac{8L^2}{(N_\gamma+1) p_{\alpha\gamma,N}(s)}. \label{eq:TN3_bd}
	\end{align}

	For $\Tmc^{\gamma,N,4}$, we have
	\begin{align}
		\Ebf \left[\Tmc^{\gamma,N,4}(s)\right]^2 & = \Ebf \left\| \left( \sum_{j \in \Ngamma} \frac{\xi_{\ialpha j}(s)}{N_{\ialpha,\gamma}(s)} \one_{\{N_{\ialpha,\gamma}(s) > 0\}} - 1 \right) b_{\alpha\gamma}(X^{\ialpha}_s,\mu^\gamma_s) \right\|^2 \notag \\
		& \le L^2 \Pbd(N_{\ialpha,\gamma}(s) = 0)  \le L^2 (1-p_{\alpha\gamma,N}(s))^{N_\gamma}  \le L^2 e^{-N_\gamma p_{\alpha\gamma,N}(s)}. \label{eq:TN4_bd}
	\end{align}
	
	Combining \eqref{eq:NpN_2}, \eqref{eq:TN1_bd} and \eqref{eq:TN2_bd} -- \eqref{eq:TN4_bd} gives us
	\begin{align}
		& \Ebf \left\| b_{\alpha\gamma}(Z^{\ialpha,N}_s, \mu^{\ialpha,\gamma,N}_s) - b_{\alpha\gamma}(X^{\ialpha}_s, \mu^\gamma_s) \right\|^2 \notag \\
		& \quad \le \kappa_3 \max_{i \in \Nbd} \Ebf \left\|Z^{i,N} - X^i\right\|_{*,s}^2 + \kappa_3 \left(\frac{1}{N_\alpha p_{\alpha\gamma,N}(s)} + e^{-N_\gamma p_{\alpha\gamma,N}(s)}\right)^{1/2} + \frac{\kappa_3}{N_\gamma p_{\alpha\gamma,N}(s)} + \kappa_3 e^{-N_\gamma p_{\alpha\gamma,N}(s)}. \label{eq:NpN_3}
	\end{align}
	From exactly the same argument as above, it is clear that \eqref{eq:NpN_3} holds with $b_{\alpha\gamma}$ replaced by $\sigma_{\alpha\gamma}$, for all $\alpha, \gamma \in \Kbd$.
	So we have from \eqref{eq:NpN_1} that
	\begin{align*}
		\max_{i \in \Nbd} \Ebf \left\|Z^{i,N} - X^i\right\|_{*,t}^2
%		& \le \int_0^t \Ebf \sum_{n=1}^4 \Tmc^{N,n}(s) \, ds \\
		& \le \kappa_4 \int_0^t \max_{i \in \Nbd} \Ebf \left\|Z^{i,N} - X^i\right\|_{*,s}^2 \, ds + \kappa_4 \left(\frac{1}{\Nbar \pbar_N} + e^{-\Nbar \pbar_N}\right)^{1/2} \\
		& \quad + \frac{\kappa_4}{\Nbar \pbar_N} + \kappa_{4} e^{-\Nbar \pbar_N}.
	\end{align*}
	The result now follows from Gronwall's lemma. \qed

	\begin{remark}
		\label{rem:remsec6}
		With a slight modification to the proof it can be shown that
		if one assumes diffusion coefficients $\{\sigma_{\alpha\gamma}\}_{\alpha,\gamma \in \Kbd}$ to be constants, then the estimate in \eqref{eq:NpN_better} holds.
		To see this,  first note that \eqref{eq:NpN_1}--\eqref{eq:TN2_alpha} are still valid with each second moment replaced by first moment.
		Combining \eqref{eq:TN2_gamma} and \eqref{eq:TN2_alpha}, using Cauchy-Schwarz inequality as in \eqref{eq:TN2_temp}, applying Lemma \ref{lem:prep_2} and \eqref{eq:ZiXi_bd}, one can argue \eqref{eq:TN2_bd} holds with second moment replaced by first moment.
	%	in the proof of Theorem \ref{thm:NpN_rate}, the leading order $1/\sqrt{\Nbar \pbar_N}$ comes from the first term on the right hand side of \eqref{eq:TN2_temp}.
	%	The proof of \eqref{eq:NpN_better} follows along the same line as that of Theorem \ref{thm:NpN_rate}, with each second moment in \eqref{eq:NpN_1}--\eqref{eq:TN2_temp} replaced by first moment.
		Also one can apply Cauchy-Schwarz inequality to get estimate of first moment from \eqref{eq:TN3_bd} and \eqref{eq:TN4_bd}.
		The desired result then follows once more from an application of Gronwall's lemma.	
	\end{remark}

%----------------------------------------------------------






\section{Proof of Theorem \ref{thm:CLT}} \label{sec:pf_CLT}
In this section we prove Theorem \ref{thm:CLT}. Conditions \ref{cond:coefficients} and \ref{cond:cond2}
will be in force throughout the section and thus will not be noted explicitly in the statement of various results.
For $N \in \Nmb$, let $\Omega_N, \Pmb^N, V_*, V^i, i \in \Nbd, \nu$ be as in Section \ref{sec:canonical_processes}.
For $t \in [0,T]$ and $i \in \Nbd$, define 
%$\{J^N(t)\}$ and $\mutil^N_t$ as
\begin{gather*}
	J^N(t) \doteq J^{N,1}(t) - \frac{1}{2} J^{N,2}(t), \quad \mutil^{i,N}_t \doteq \frac{1}{N_i(t)} \sum_{j=1}^N \xi_{ij}(t) \delta_{X^j_t},
\end{gather*}
where
\begin{equation} \label{eq:JN1}
	J^{N,1}(t) \doteq \sum_{i=1}^N \int_0^t \left( b(X^i_s,\mutil^{i,N}_s) - b(X^i_s,\mu_s) \right) \cdot d W^i_s
\end{equation}
and
\begin{equation} \label{eq:JN2}
	J^{N,2}(t) \doteq \sum_{i = 1}^N \int_0^t \left\| b(X^i_s,\mutil^{i,N}_s) - b(X^i_s,\mu_s) \right\|^2 \, ds.
\end{equation}
Let $\Fmctil^N_t \doteq \sigma\{V^i(s), \xi_{ij}(s),  0 \le s \le t, i, j \in \Nbd \}$.
Note that $\left\{\exp\left(J^N(t)\right)\right\}$ is an $\{\Fmctil^N_t\}$-martingale under $\Pmb^N$.
Define a new probability measure $\Qmb^N$ on $\Omega_N$ by 
\begin{equation*}
	\frac{d\Qmb^N}{d\Pmb^N} \doteq \exp\left(J^N(T)\right).
\end{equation*}
%Expected value under $\Qmb^N$ will be denoted as $\Ebf_{\Qmb^N}$.

By Girsanov's Theorem, $(X^1, \dotsc, X^N, \{\xi_{ij} : i,j \in \Nbd \})$ has the same probability distribution under $\Qmb^N$ as $(Z^{1,N}, \dotsc, Z^{N,N}, \{\xi_{ij} : i,j \in \Nbd \})$ under $\Pbd$.
For $\phi \in L^2_c(\Cmc_d,\mu)$, let 
\begin{align}\label{eq:eq1231}\etatil^N (\phi) \doteq \frac{1}{\sqrt{N}} \sum_{i = 1}^N \phi (X^i).
\end{align}
Thus in order to prove the theorem it suffices to show that for any $\phi \in L^2_c(\Cmc_d,\mu)$,
\begin{equation*}
	\lim_{N \to \infty} \Ebf_{\Qmb^N} \exp \left( i \etatil^N(\phi) \right) = \exp \left( -\frac{1}{2} \left\| (I-A)^{-1} \phibd \right\|^2_{L^2(\Omega_d,\nu)} \right),
\end{equation*}
which is equivalent to showing
\begin{equation} \label{eq:cf_cvg}
	\lim_{N \to \infty} \Ebf_{\Pmb^N} \exp \left( i \etatil^N(\phi) + J^{N,1}(T) - \frac{1}{2} J^{N,2}(T) \right) = \exp \left( -\frac{1}{2} \left\| (I-A)^{-1} \phibd \right\|^2_{L^2(\Omega_d,\nu)} \right).
\end{equation}
For this we will need to study the asymptotics of $J^{N,1}$ and $J^{N,2}$ as $N \to \infty$. 



%----------------------------------------------------------




\subsection{Asymptotics of symmetric statistics} \label{sec:asymp_symmetric_statistics}
The proof of \eqref{eq:cf_cvg} crucially relies on certain classical results from \cite{Dynkin1983} on limit laws of degenerate symmetric statistics.
In this section we briefly review these results.

Let $\Smb$ be a Polish space and let $\{Y_n\}_{n=1}^\infty$ be a sequence of i.i.d.\ $\Smb$-valued random variables having common probability law $\theta$.
For $k \in \Nmb$, let $L^2(\theta^{\otimes k})$ be the space of all real-valued square integrable functions on $(\Smb^k, \Bmc(\Smb)^{\otimes k}, \theta^{\otimes k})$.
Denote by $L^2_c(\theta^{\otimes k})$ the subspace of centered functions, namely $\phi \in L^2(\theta^{\otimes k})$ such that for all $1 \le j \le k$,
\begin{equation*}
	\int_\Smb \phi(x_1,\dotsc,x_{j-1},x,x_{j+1},\dotsc,x_k) \, \theta(dx) = 0, \quad \theta^{\otimes k-1} \text{ a.e. } (x_1,\dotsc,x_{j-1},x_{j+1},\dotsc,x_k).
\end{equation*}
Denote by $L^2_{sym}(\theta^{\otimes k})$ the subspace of symmetric functions, namely $\phi \in L^2(\theta^{\otimes k})$ such that for every permutation $\pi$ on $\{1,\dotsc,k\}$,
\begin{equation*}
	\phi(x_1,\dotsc,x_k) = \phi(x_{\pi(1)},\dotsc,x_{\pi(k)}), \quad \theta^{\otimes k} \text{ a.e. } (x_1,\dotsc,x_k).
\end{equation*}
Also, denote by $L^2_{c,sym}(\theta^{\otimes k})$ the subspace of centered symmetric functions in $L^2(\theta^{\otimes k})$, namely $L^2_{c,sym}(\theta^{\otimes k}) \doteq L^2_c(\theta^{\otimes k}) \bigcap L^2_{sym}(\theta^{\otimes k})$.
Given $\phi_k \in L^2_{sym}(\theta^{\otimes k})$ define the symmetric statistic $\Umc^n_k (\phi_k)$ as
\begin{equation*}
	\Umc^n_k (\phi_k) \doteq
	\begin{cases}
		\displaystyle \sum_{1 \le i_1 < i_2 < \dotsb < i_k \le n} \phi_k(Y_{i_1},\dotsc,Y_{i_k}) & \text{for } n \ge k \\
		0 & \text{for } n<k.
	\end{cases}
\end{equation*}
In order to describe the asymptotic distributions of such statistics consider a Gaussian field $\{I_1(h) : h \in L^2(\theta)\}$ such that
\begin{equation*}
	\Ebf \left( I_1(h) \right) = 0, \: \Ebf \left( I_1(h)I_1(g) \right) = \langle h,g \rangle_{L^2(\theta)}, \quad h,g \in L^2(\theta).
\end{equation*}
For $h \in L^2(\theta)$, define $\phi_k^h \in L^2_{sym}(\theta^{\otimes k})$ as
\begin{equation*}
	\phi_k^h(x_1,\dotsc,x_k) \doteq h(x_1) \dotsc h(x_k)
\end{equation*}
and set $\phi_0^h \doteq 1$.

The multiple Wiener integral (MWI) of $\phi_k^h$, denoted as $I_k(\phi_k^h)$, is defined through the following formula.
For $k \ge 1$,
\begin{equation*} 
%\label{eq:MWI_formula}
	I_k(\phi_k^h) \doteq \sum_{j=0}^{\lfloor k/2 \rfloor} (-1)^j C_{k,j} ||h||^{2j}_{L^2(\theta)} (I_1(h))^{k-2j}, \text{ where } C_{k,j} \doteq \frac{k!}{(k-2j)! 2^j j!}, j=0,\dotsc,\lfloor k/2 \rfloor.
\end{equation*}
The following representation gives an equivalent way to characterize the MWI of $\phi_k^h $:
\begin{equation*}
	\sum_{k=0}^{\infty} \frac{t^k}{k!} I_k(\phi_k^h) = \exp \left( tI_1(h) - \frac{t^2}{2} ||h||^2_{L^2(\theta)} \right), \quad t \in \Rmb,
\end{equation*}
where we set $I_0(\phi_0^h) \doteq 1$. 
We extend the definition of $I_k$ to the linear span of $\{\phi_k^h, h \in L^2(\theta)\}$ by linearity. 
It can be checked that for all $f$ in this linear span,
\begin{equation} \label{eq:MWI_moments}
	\Ebf(I_k(f))^2 = k! \, ||f||^2_{L^2(\theta^{\otimes k})}.
\end{equation}
Using this identity and standard denseness arguments, the definition of $I_k(f)$ can be extended to all $f \in L^2_{sym}(\theta^{\otimes k})$ and the identity $\eqref{eq:MWI_moments}$ holds for all $f \in L^2_{sym}(\theta^{\otimes k})$. 
The following theorem is taken from \cite{Dynkin1983}.

\begin{theorem}[Dynkin-Mandelbaum \cite{Dynkin1983}] \label{thm:Dynkin}
	Let $\{ \phi_k \}_{k=1}^\infty$ be such that $\phi_k \in L^2_{c,sym}(\theta^{\otimes k})$ for each $k \ge 1$. 
	Then the following convergence holds as $n \to \infty$:
	\begin{equation*}
		\left( n^{-\frac{k}{2}} \Umc^n_k (\phi_k) \right)_{k \ge 1} \Rightarrow \left( \frac{1}{k!} I_k(\phi_k) \right)_{k \ge 1}
	\end{equation*}
	as a sequence of $\Rmb^\infty$-valued random variables.
\end{theorem}

%Now let $\{\zeta_{ij} : 1 \le i < j \le n\}$ be i.i.d.\ Bernoulli($p_n$) random variables, which are also independent of $\{Y_i : i \in \Nbd\}$.
%Let $f \in L^2_{c,sym}(\theta^{\otimes 2})$ and
%\begin{equation*}
%	U_n = \frac{1}{\sqrt{\binom{n}{2}p_n}} \sum_{1 \le i < j \le n} (\zeta_{ij}-p_n) f(Y_i,Y_j).
%\end{equation*}
%The following theorem is taken from \cite{Janson1984}.
%
%\begin{theorem}[Janson \cite{Janson1984}] \label{thm:Janson}
%	Let $f$ and $U_n$ be as above.
%	Suppose $p_n \to p \in [0,1]$ as $n \to \infty$.
%	Let $\alpha^2 = (1-p)\|f\|_{L^2(\theta^{\otimes 2})}^2$.
%	Then as $n \to \infty$,
%	\begin{equation*}
%		\left( U_n, \frac{1}{n}\sigma^n_2(f) \right) \Rightarrow \left( Z, \frac{1}{2} I_2(f) \right),
%	\end{equation*}
%	where $Z \sim N(0,\alpha^2)$ is independent of $I_2(f)$.
%\end{theorem}
%
%Similar to above theorem, one can show the following lemma, which will be used in studying asymptotics of $J^{N,1}$.
%
%\begin{lemma} \label{lem:incompletestat}
%	Let $\{\zeta_{ij}(t) : 1 \le i < j \le n\}$ be i.i.d.\ RCLL processes with marginal Bernoulli($p_n(t)$) distributions, which are also independent of $\{Y_i : i \in \Nbd\}$.
%	Let $h$ be such that for each $\omega \in \Dmb_{\{0,1\}}[0,T]$ $h(\omega,\cdot,\cdot) \in L^2_{c,sym}(\theta^{\otimes 2})$ and for $\theta^{\otimes 2}$ a.e.\ $(x,y) \in \Smb^2$, $\Ebf(h(\zeta_{ij},x,y)) = 0$.
%	Let
%	\begin{equation*}
%		V_n = \frac{1}{\sqrt{\binom{n}{2}}} \sum_{1 \le i < j \le n} h(\zeta_{ij},Y_i,Y_j).
%	\end{equation*}
%	Suppose $p_n \to p \in \Dmb_{\{0,1\}}[0,T]$ as $n \to \infty$.
%	Let $\alpha^2 = \Ebf(h^2(\zeta,Y_1,Y_2))$ where $\zeta$ is RCLL process with marginal Bernoulli($p(t)$) distributions and independent of $Y_1$ and $Y_2$.
%	Then as $n \to \infty$,
%	\begin{equation*}
%		\left(V_n, \left( n^{-\frac{k}{2}} \sigma^n_k (\phi_k) \right)_{k \ge 1} \right) \Rightarrow \left(Z, \left(\frac{1}{k!} I_k(\phi_k) \right)_{k \ge 1} \right)
%	\end{equation*}
%	as a sequence of $\Rmb^\infty$-valued random variables, where $Z \sim N(0,\alpha^2)$ is independent of all $I_k(\phi_k)$. 	
%\end{lemma}




\subsection{Completing the proof of Theorem \ref{thm:CLT}} \label{sec:complete_pf_CLT}

From Condition \ref{cond:cond2} for any subsequence of $\{p_N\}$ there is a further subsequence and a $p \in \Cmb([0,T]:\Rmb)$ such that, along the subsequence, $p_N$ converges to $p$.
In order to prove Theorem \ref{thm:CLT} it suffices to prove the statement in the theorem along every such subsequence. Thus, without loss of generality, we will assume that for some
$p \in \Cmb([0,T]:\Rmb)$,
\begin{equation}
	\label{eq:eq416}
	p_N \to p,\; \bar p \doteq \inf_{0\le s \le T} p(s) >0 \mbox{ and } p(s) \in [0,1] \mbox{ for all } s \in [0,T].
\end{equation}

Let $(\Omega^*,\Fmc^*, \Pbd^*)$ be some probability space on which we are given
\begin{enumerate}[$(a)$]
\item
	 a Gaussian random field $\{I_1(h): h \in L^2(\nu)\}$ as in Section \ref{sec:asymp_symmetric_statistics} with $\Smb = \Omega_d$ and $\theta = \nu$ (see Section \ref{sec:canonical_processes}).
\item
	a Gaussian random variable $Z$ with  zero mean and variance
	\begin{equation} \label{eq:sigmasq}
		\sigma^2 \doteq \int_0^T \frac{1-p(s)}{p(s)} \lambda_s \, ds,
	\end{equation}
	\end{enumerate}
	where $\lambda_t$ is defined in \eqref{eq:lambdas}, such that
	$Z$ is independent of $\{I_1(h): h \in L^2(\nu)\}$. Note that \eqref{eq:eq416} implies $\sigma^2 < \infty$.
	Define on this probability space multiple Wiener integrals $I_k(f)$ for $k \ge 1$ and $f \in L^2_{sym}(\nu^{\otimes k})$
	as in Section \ref{sec:asymp_symmetric_statistics}. Note that the collection
	$\{I_k(f): k\ge 1, f \in L^2_{sym}(\nu^{\otimes k})\}$ is independent of $Z$.
Recall functions $h$ and $\phibd$ (corresponding to each $\phi \in L^2_c(\Cmc_d,\mu)$) defined in Sections \ref{sec:some_integral_operators} and \ref{sec:CLT}. 
Define 
\begin{align}
	h^{sym}(\omega,\omega') & \doteq \half(h(\omega,\omega') + h(\omega',\omega)), \quad (\omega,\omega') \in \Omega_d^2 \label{eq:h_sym} \\
	m_t(x,y) & \doteq \int_{\Rmb^d} \bbar_t(z,x) \cdot \bbar_t(z,y) \, \mu_t(dz), \quad x,y \in \Rd, t \in [0,T] \label{eq:m_t} \\
	l(\omega,\omega') & \doteq \int_0^T m_s(X_{*,s}(\omega),X_{*,s}(\omega')) \, ds, \quad (\omega,\omega') \in \Omega_d^2. \label{eq:l}
\end{align}
In order to prove \eqref{eq:cf_cvg} (which will complete the proof of Theorem \ref{thm:CLT}), we will make use of the following key proposition, whose proof will be given in Section \ref{sec:pf_prop_key}. Recall $J^{N,1}$ and $J^{N,2}$
defined in \eqref{eq:JN1} and \eqref{eq:JN2} respectively. Also recall $\etatil^N(\phi)$ associated with
a $\phi \in  L^2_c(\Cmc_d,\mu)$ introduced in \eqref{eq:eq1231}.

\begin{proposition}
	\label{prop:key_joint_cvg}
	Let $\phi \in  L^2_c(\Cmc_d,\mu)$. Then,
	as $N \to \infty$,
	\begin{equation}
		\label{eq:joint_cvg}
		(\etatil^N(\phi), J^{N,1}(T), J^{N,2}(T)) \Rightarrow \left(I_1(\phibd), Z + I_2(h^{sym}), I_2(l) + \int_0^T \frac{1}{p(s)} \lambda_s \, ds \right).
	\end{equation}
\end{proposition}
We now complete the proof of \eqref{eq:cf_cvg} using the above proposition.
It follows from the  proposition that as $N \to \infty$,
\begin{equation}
	\left(\etatil^N(\phi), J^N(T)\right) \Rightarrow \left(I_1(\phibd), \frac{1}{2} I_2(f) + Z - \int_0^T \frac{1}{2p(s)} \lambda_s \, ds\right) \doteq (I_1(\phibd), J),\label{eq:eq1250}
\end{equation}
where $f$ is defined as
\begin{equation*}
	f(\omega,\omega') \doteq h(\omega,\omega') + h(\omega',\omega) - l(\omega,\omega'), \quad (\omega,\omega') \in \Omega_d^2.
\end{equation*}
%and $Z$ is independent of $I_2(f)$.
Recalling the formula for $\textnormal{Trace}(AA^*)$ established in Lemma \ref{lem:trace}, the definition of
$\sigma^2$ in \eqref{eq:sigmasq}, and using independence between $Z$ and $\{I_k(\cdot)\}_{k \ge 1}$, we have
\begin{equation*}
	\Ebf_{\Pbd^*} \exp(J) = \Ebf_{\Pbd^*} \exp \left( \frac{1}{2} I_2(f) - \frac{1}{2} \textnormal{Trace}(AA^*) \right) \Ebf_{\Pbd^*} \exp \left( Z - \half \sigma^2 \right) = 1,
\end{equation*}
where the last equality follows from Lemma \ref{lem:Shiga_Tanaka} in Appendix \ref{sec:restating}.
Since $\EbfP \exp(J^N(T)) = 1$ for all $N$, we have from Scheffe's lemma that $\{\exp(J^N(T))\}$ is uniformly integrable and consequently so is $\{\exp(i\etatil^N(\phi) + J^N(T))\}$.
Hence, from \eqref{eq:eq1250}
  we have that as $N \to \infty$,
\begin{align*}
	& \quad \EbfP \exp \left( i\etatil^N(\phi) +J^N(T) \right) \\
	& \to \Ebf_{\Pbd^*} \exp \left(iI_1(\phibd) + J \right) \\
	& = \Ebf_{\Pbd^*} \exp \left( iI_1(\phibd) + \frac{1}{2} I_2(f) - \frac{1}{2} \textnormal{Trace}(AA^*) \right) \Ebf_{\Pbd^*} \exp \left( Z - \half \sigma^2 \right) \\
	& = \exp \left( -\frac{1}{2} \left\| (I-A)^{-1} \phibd \right\|^2_{L^2(\Omega_d,\nu)} \right),
\end{align*}
where the first equality uses the independence between $Z$ and $\{I_k(\cdot)\}_{k \ge 1}$ and the last equality again follows from Lemma \ref{lem:Shiga_Tanaka}.
Thus we have shown \eqref{eq:cf_cvg}, which completes the proof of Theorem \ref{thm:CLT}. \qed









%-----------------------------------------------------------------------------




\subsection{Proof of Proposition \ref{prop:key_joint_cvg}} \label{sec:pf_prop_key}

In this section we prove  Proposition \ref{prop:key_joint_cvg}.
We will first reduce $J^{N,1}(T)$ and $J^{N,2}(T)$ to forms that are more convenient to analyze.

\subsubsection{Reducing $J^{N,1}$ and $J^{N,2}$}
\label{sec:reducing}

The term $J^{N,1}(T)$ in \eqref{eq:JN1} can be written as
\begin{align*}
	J^{N,1}(T) & = \sum_{i=1}^N \int_0^T \left( b(X^i_s,\mutil^{i,N}_s) - b(X^i_s,\mu_s) \right) \cdot d W^i_s \\
	& = \sum_{i=1}^N \int_0^T \left( \frac{1}{N_i(s)} \sum_{j=1}^N \xi_{ij}(s) \bbar(X^i_s,X^j_s) - b(X^i_s,\mu_s) \right) \cdot d W^i_s \\
	& = \sum_{i=1}^N \sum_{j=1}^N \int_0^T \frac{\xi_{ij}(s)}{N_i(s)} \bbar_s(X^i_s,X^j_s) \cdot d W^i_s,
\end{align*}
where $\bbar_s$ is defined in \eqref{eq:bbar t}.
Let
\begin{equation} \label{eq:JN1til}
	\Jtil^{N,1}(T) \doteq \sum_{i=1}^N \sum_{j=1}^N \int_0^T \frac{\xi_{ij}(s)}{N p_N(s)} \bbar_s(X^i_s,X^j_s) \cdot d W^i_s.
\end{equation}

We will argue in Lemma \ref{lem:JN1_JN1_tilde} that the asymptotic behavior of $J^{N,1}$ is the same as that of $\Jtil^{N,1}(T)$, the proof of which relies on the following lemma.

\begin{lemma} \label{lem:approxNi}
	As $N \to \infty$,
	\begin{equation*}
		\sup_{s \in [0,T]} \EbfP \left[ \frac{N^2}{N_1^2(s)} - \frac{1}{p_N^2(s)} \right]^2 \to 0, \quad \sup_{s \in [0,T]} \EbfP \left[ \frac{N}{N_1(s)} - \frac{1}{p_N(s)} \right]^2 \to 0.
	\end{equation*}
\end{lemma}

\begin{proof}
	It suffices to prove the first convergence, since the second one follows from the inequality
	\begin{equation*}
		\left|\frac{N}{N_1(s)} - \frac{1}{p_N(s)}\right| = \frac{\left|\frac{N^2}{N_1^2(s)} - \frac{1}{p_N^2(s)}\right|}{\left|\frac{N}{N_1(s)} + \frac{1}{p_N(s)}\right|} \le \left|\frac{N^2}{N_1^2(s)} - \frac{1}{p_N^2(s)}\right|.
	\end{equation*}
	Recall $C_N(\cdot)$ from Lemma \ref{lem:prep_4}.
	For $s \in [0,T]$, consider the event $G_N(s) \doteq \{ \omega \in \Omega_N : |N_1(s) - Np_N(s)| > C_N(3) + 1 \}$.
	It follows from Lemma \ref{lem:prep_4} that $\Pmb^N(G_N(s)) \le \frac{2}{N^6}$.
	Write
	\begin{equation} \label{eq:approxNi}
		\left[ \frac{N^2}{N_1^2(s)} - \frac{1}{p_N^2(s)} \right]^2 = \frac{(N_1^2(s) - N^2p_N^2(s))^2}{N_1^4(s) p_N^4(s)} \one_{G_N(s)} + \frac{(N_1^2(s) - N^2p_N^2(s))^2}{N_1^4(s) p_N^4(s)} \one_{G_N^c(s)}.
	\end{equation}
	Noting that $| N_1^2(s) - N^2 p_N^2(s)| \le N^2$ and $N_1(s) \ge 1$, we have as $N \to \infty$,
	\begin{equation} \label{eq:approxNi1}
		\EbfP \left[\frac{(N_1^2(s) - N^2p_N^2(s))^2}{N_1^4(s) p_N^4(s)} \one_{G_N(s)}\right] \le \frac{N^4}{p_N^4(s)} \Pmb^N(G_N(s)) \le \frac{2}{N^2 \pbar_N^4} \to 0,
	\end{equation}
	where the convergence follows from Condition \ref{cond:cond2}.
	Now consider the second term on the right side of \eqref{eq:approxNi}.
	Condition \ref{cond:cond2} implies that $N \pbar_N - C_N(3) - 1 > 0$ for large enough $N$.
	Hence
	\begin{equation} \label{eq:approxNi2}
		\begin{aligned} 
			\EbfP \left[\frac{(N_1^2(s) - N^2p_N^2(s))^2}{N_1^4(s) p_N^4(s)} \one_{G_N^c(s)}\right] & \le \EbfP \left[\frac{(C_N(3)+1)^2 (N_1(s)+Np_N(s))^2}{(Np_N(s) - C_N(3) - 1)^4p_N^4(s)} \one_{G_N^c(s)}\right] \\
			& \le \frac{4N^2(\sqrt{3 (N-1) \log N} + 1)^2}{(N \pbar_N - \sqrt{3 (N-1) \log N} - 1)^4 \pbar_N^4} \to 0
		\end{aligned}		
	\end{equation}
	as $N \to \infty$.
	The result follows by combining \eqref{eq:approxNi1} and \eqref{eq:approxNi2}.
\end{proof}

The following lemma says that to study the asymptotics of $J^{N,1}(T)$, it suffices to study the asymptotic behavior of $\Jtil^{N,1}(T)$.

\begin{lemma} \label{lem:JN1_JN1_tilde}
	\begin{equation*}
		\lim_{N \to \infty} \EbfP \left|J^{N,1}(T) - \Jtil^{N,1}(T)\right|^2 = 0.
	\end{equation*}
\end{lemma}

\begin{proof}
	First note that as $N \to \infty$
	\begin{align*}
		& \EbfP \left| \sum_{1 \le i < j \le N} \int_0^T \frac{\xi_{ij}(s)}{N_i(s)} \bbar_s(X^i_s,X^j_s) \cdot d W^i_s - \sum_{1 \le i < j \le N} \int_0^T \frac{\xi_{ij}(s)}{N p_N(s)} \bbar_s(X^i_s,X^j_s) \cdot d W^i_s \right|^2 \\
		& \quad = \EbfP \left( \sum_{1 \le i < j \le N} \int_0^T \left(\frac{1}{N_i(s)} - \frac{1}{Np_N(s)}\right) \xi_{ij}(s) \bbar_s(X^i_s,X^j_s) \cdot d W^i_s \right)^2 \\
		& \quad = \sum_{1 \le i < j \le N} \int_0^T \left( \EbfP \left[\left(\frac{1}{N_i(s)} - \frac{1}{Np_N(s)}\right) \xi_{ij}(s)\right]^2 \EbfP \bbar_s^2(X^i_s,X^j_s) \right) \, ds \\
		& \quad \le \kappa N^2 \int_0^T \EbfP \left(\frac{1}{N_1(s)} - \frac{1}{Np_N(s)}\right)^2 \, ds \to 0,
%		& = \kappa \EbfP \frac{(N_1 - Np_N)^2}{N_1^2 p_N^2} \\
	\end{align*}
	where the convergence follows from Lemma \ref{lem:approxNi}.
	Similarly one can show that
	\begin{gather*}
		\lim_{N \to \infty} \EbfP \left| \sum_{1 \le j < i \le N} \int_0^T \frac{\xi_{ij}(s)}{N_i(s)} \bbar_s(X^i_s,X^j_s) \cdot d W^i_s - \sum_{1 \le j < i \le N} \int_0^T \frac{\xi_{ij}(s)}{N p_N(s)} \bbar_s(X^i_s,X^j_s) \cdot d W^i_s \right|^2 = 0, \\
		\lim_{N \to \infty} \EbfP \left| \sum_{i=1}^N \int_0^T \frac{1}{N_i(s)} \bbar_s(X^i_s,X^i_s) \cdot d W^i_s - \sum_{i=1}^N \int_0^T \frac{1}{N p_N(s)} \bbar_s(X^i_s,X^i_s) \cdot d W^i_s \right|^2 = 0.
	\end{gather*} 
	Combining above results completes the proof.
\end{proof}




%%----------------------------------------------------------
%
%
%
%\subsubsection{Reducing $J^{N,2}$} \label{sec:asymp_JN2}

Next we study the asymptotics of $J^{N,2}(T)$. For that we will need the following lemma.

\begin{lemma} \label{lem:Ttil}
	Suppose $\gamma, \vartheta, \rho$ are bounded measurable real maps on $[0,T]\times (\Rmb^d)^3$,
	$[0,T]\times (\Rmb^d)^2$ and $[0,T]\times \Rmb^d$ respectively, such that for all $s\in [0,T]$ 
 $\gamma_s \equiv \gamma(s, \cdot) \in L^2_c(\mu_s^{\otimes 3})$, $\vartheta_s \equiv \vartheta(s, \cdot)\in L^2_c(\mu_s^{\otimes 2})$ and $\rho_s \equiv \rho(s, \cdot)\in L^2_c(\mu_s)$.
	Then as $N \to \infty$
	\begin{align}
		& \EbfP \left| \sum_{i\neq j, j\neq k, i\neq k} \int_0^T \frac{\xi_{ij}(s)\xi_{ik}(s)}{N_i^2(s)} \gamma_s(X^i_s,X^j_s,X^k_s) \, ds \right| \to 0, \label{eq:Ttilijk}\\
		& \EbfP \left| \sum_{1 \le i \ne k \le N} \int_0^T \frac{\xi_{ik}(s)}{N_i^2(s)} \vartheta_s(X^i_s,X^k_s) \, ds \right| \to 0, \label{eq:Ttilik}\\
		& \EbfP \left| \sum_{1 \le i \ne k \le N} \int_0^T \frac{\xi_{ik}(s)}{N_i^2(s)} \rho_s(X^k_s) \, ds \right| \to 0, \label{eq:Ttilk} \\
		& \EbfP \left| \sum_{1 \le i \ne k \le N} \int_0^T \frac{\xi_{ik}(s)}{N_i^2(s)} \rho_s(X^i_s) \, ds \right| \to 0 \label{eq:Ttili}.
	\end{align}
\end{lemma}

\begin{proof}
	To prove \eqref{eq:Ttilijk}, it is enough to prove the convergence with the summation taken over the ordered sum $i < j < k$.
	Now note that
	\begin{align*}
		& \EbfP \left( \sum_{1 \le i < j < k \le N} \int_0^T \frac{\xi_{ij}(s)\xi_{ik}(s)}{N_i^2(s)} \gamma_s(X^i_s,X^j_s,X^k_s) \, ds \right)^2 \\
		& \quad = \EbfP \sum_{1 \le i < j < k \le N} \left( \int_0^T \frac{\xi_{ij}(s)\xi_{ik}(s)}{N_i^2(s)} \gamma_s(X^i_s,X^j_s,X^k_s) \, ds \right)^2 \\
		& \quad \le \kappa_1 N^3 \int_0^T \EbfP \frac{1}{N_1^4(s)} \, ds \le \kappa_2 N^3 \int_0^T \frac{1}{N^4 p_N^4(s)} \, ds \le \frac{\kappa_2 T}{N \pbar_N^4} \to 0,
	\end{align*}
	where the second inequality follows from Lemma~\ref{lem:prep_1}.
	Thus \eqref{eq:Ttilijk} holds.
	Proofs for \eqref{eq:Ttilik}, \eqref{eq:Ttilk} and \eqref{eq:Ttili} are similar and hence omitted.
\end{proof}
Recall the definition of $m_t$ from \eqref{eq:m_t}. Define
\begin{equation}\label{eq:eq114}
	\Jtil^{N,2}(T) \doteq \frac{N-2}{N^2}\sum_{1 \le j\neq k \le N} \int_0^T m_s(X^j_s,X^k_s) \, ds + \int_0^T \frac{\lambda_s}{p(s)}  \, ds.
\end{equation}
The following lemma shows that $\Jtil^{N,2}(T)$ is asymptotically the same as $J^{N,2}(T)$.
\begin{lemma}
	\label{lem:tiljn2}
	As $N\to \infty$, $J^{N,2}(T)-\Jtil^{N,2}(T)$ converges to $0$ in probability.
\end{lemma}
\begin{proof}
	%Next recall the definition of $J^{N,2}$ in \eqref{eq:JN2}.
We split $J^{N,2}(T)$ as follows:
\begin{align*}
	J^{N,2}(T) & 
%	\sum_{i = 1}^N \int_0^T \left\| b(X^i_s,\mu^{i,N}_s) - b(X^i_s,\mu_s) \right\|^2 \, ds \\
%	& = \sum_{i = 1}^N \int_0^T \left\| \frac{1}{N_i(s)} \sum_{j=1}^N \xi_{ij(s)} \bbar(X^i_s,X^j_s) - b(X^i_s,\mu_s) \right\|^2 \, ds \\
	= \sum_{i,j,k=1}^N \int_0^T \frac{\xi_{ij}(s) \xi_{ik}(s)}{N_i^2(s)} \bbar_s(X^i_s,X^j_s) \cdot \bbar_s(X^i_s,X^k_s) \, ds \\
	& = \sum_{n=1}^5 \sum_{(i,j,k) \in \Smc_n} \int_0^T \frac{\xi_{ij(s)} \xi_{ik}(s)}{N_i^2(s)} \bbar_s(X^i_s,X^j_s) \cdot \bbar_s(X^i_s,X^k_s) \, ds \doteq \sum_{n=1}^5 \Tmctil^{N,n},
\end{align*}
where $\Smc_1$, $\Smc_2$, $\Smc_3$, $\Smc_4$ and $\Smc_5$ are collections of $(i,j,k) \in \Nbd^3$ such that $\{i=j=k\}$, $\{i=j \ne k\}$, $\{i=k \ne j\}$, $\{j=k \ne i\}$ and $\{i,j,k \text{ distinct}\}$, respectively.
%\begin{gather*}
%	\Smc_1 = \{ (i,j,k) \in \Nbd^3: 1 \le i,j,k \le N, i=j=k \}, \\
%	\Smc_2 = \{ (i,j,k) \in \Nbd^3: 1 \le i,j,k \le N, i=j \ne k \}, \\
%	\Smc_3 = \{ (i,j,k) \in \Nbd^3: 1 \le i,j,k \le N, i=k \ne j \}, \\
%	\Smc_4 = \{ (i,j,k) \in \Nbd^3: 1 \le i,j,k \le N, j=k \ne i \}, \\
%	\Smc_5 = \{ (i,j,k) \in \Nbd^3: 1 \le i,j,k \le N, i,j,k \text{ distinct} \}.
%\end{gather*}
For $\Tmctil^{N,1}$, we have
\begin{align*}
	\EbfP | \Tmctil^{N,1} | & = \EbfP \sum_{i=1}^N \int_0^T \frac{1}{N_i^2(s)} \| \bbar_s(X^i_s,X^i_s) \|^2 \, ds \le \kappa_1 N \int_0^T \EbfP \frac{1}{N_1^2(s)} \, ds \\
	& \le \kappa_2 N \int_0^T \frac{1}{N^2 p_N^2(s)} \, ds \le \frac{\kappa_2 N T}{N^2 \pbar_N^2} \to 0
\end{align*}
as $N \to \infty$, where the second inequality follows from Lemma \ref{lem:prep_1}.	
To study the asymptotics of $\Tmctil^{N,2}$, $\Tmctil^{N,3}$, $\Tmctil^{N,4}$ and $\Tmctil^{N,5}$, we will use Lemma \ref{lem:Ttil}.

For $\Tmctil^{N,2}$, note that
\begin{align*}
	\Tmctil^{N,2} & = \sum_{1 \le i \ne k \le N} \int_0^T \frac{\xi_{ik}(s)}{N_i^2(s)} \bbar_s(X^i_s,X^i_s) \cdot \bbar_s(X^i_s,X^k_s) \, ds \\
	& = \sum_{1 \le i \ne k \le N} \int_0^T \frac{\xi_{ik}(s)}{N_i^2(s)} \left( \bbar_s(X^i_s,X^i_s) \cdot \bbar_s(X^i_s,X^k_s) - \intRd \bbar_s(y,y) \cdot \bbar_s(y,X^k_s) \, \mu_s(dy) \right) \, ds \\
	& \quad + \sum_{1 \le i \ne k \le N} \int_0^T \frac{\xi_{ik}(s)}{N_i^2(s)} \intRd \bbar_s(y,y) \cdot \bbar_s(y,X^k_s) \, \mu_s(dy) \, ds.
\end{align*}
It then follows from Lemma \ref{lem:Ttil} (see \eqref{eq:Ttilik} and \eqref{eq:Ttilk}) that $\EbfP |\Tmctil^{N,2}| \to 0$ as $N \to \infty$.
Similarly $\EbfP |\Tmctil^{N,3}| \to 0$ as $N \to \infty$.
	
Consider now the fourth term $\Tmctil^{N,4}$. 
Recalling $\lambda_t$ defined in \eqref{eq:lambdas}, we write
\begin{align*}
	\Tmctil^{N,4} & = \sum_{1 \le i \ne j \le N} \int_0^T \frac{\xi_{ij}(s)}{N_i^2(s)} \| \bbar_s(X^i_s,X^j_s) \|^2 \, ds \\
	& = \sum_{1 \le i \ne j \le N} \int_0^T \frac{\xi_{ij}(s)}{N_i^2(s)} \bigg( \| \bbar_s(X^i_s,X^j_s) \|^2 - \lan \|\bbar_s(X^i_s,\cdot)\|^2, \mu_s \ran - \lan \| \bbar_s(\cdot,X^j_s) \|^2,\mu_s \ran + \lambda_s \bigg) ds \\
	& \qquad + \sum_{1 \le i \ne j \le N} \int_0^T \frac{\xi_{ij}(s)}{N_i^2(s)} \left( \lan \|\bbar_s(X^i_s,\cdot)\|^2, \mu_s \ran - \lambda_s \right) ds \\
	& \qquad + \sum_{1 \le i \ne j \le N} \int_0^T \frac{\xi_{ij}(s)}{N_i^2(s)} \left( \lan \| \bbar_s(\cdot,X^j_s) \|^2,\mu_s \ran - \lambda_s \right) ds + \sum_{1 \le i \ne j \le N} \int_0^T \frac{\xi_{ij}(s)}{N_i^2(s)} \lambda_s \, ds \\
	& \doteq \sum_{n=1}^4 \Tmctil^{N,4}_n.
\end{align*}
It follows from Lemma \ref{lem:Ttil} (see \eqref{eq:Ttilik}, \eqref{eq:Ttilk} and \eqref{eq:Ttili}) that as $N \to \infty$,
\begin{equation*}
	\EbfP |\Tmctil^{N,4}_n| \to 0, \quad n=1,2,3.
\end{equation*}
We now show that
\begin{equation} \label{eq:TN44}
	\EbfP \left| \Tmctil^{N,4}_4 - \int_0^T \frac{1}{p(s)} \lambda_s \, ds \right| \to 0
\end{equation}
as $N \to \infty$.
For this, first write
\begin{equation*}
	\Tmctil^{N,4}_4 = \sum_{i=1}^N \int_0^T \frac{N_i(s)-1}{N_i^2(s)} \lambda_s \, ds = \sum_{i=1}^N \int_0^T \frac{1}{N_i(s)} \lambda_s \, ds - \sum_{i=1}^N \int_0^T \frac{1}{N_i^2(s)} \lambda_s \, ds.
\end{equation*}
By Lemma~\ref{lem:approxNi}
\begin{equation*}
	\lim_{N \to \infty} \EbfP \sum_{i=1}^N \int_0^T \frac{1}{N_i^2(s)} \lambda_s \, ds = \lim_{N \to \infty} \int_0^T \EbfP \frac{N}{N_1^2(s)} \lambda_s \, ds = 0.
\end{equation*}
Also as $N \to \infty$,
\begin{align*}
	\EbfP \left| \sum_{i=1}^N \int_0^T \frac{1}{N_i(s)} \lambda_s \, ds - \int_0^T \frac{1}{p(s)} \lambda_s \, ds \right|^2 & \le \kappa_3 N \EbfP \sum_{i=1}^N \int_0^T \left( \frac{1}{N_i(s)} - \frac{1}{Np(s)} \right)^2 \, ds \\
	& \le \kappa_3 N^2 \int_0^T \EbfP \left( \frac{1}{N_1(s)} - \frac{1}{Np(s)} \right)^2 \, ds \to 0,
\end{align*}
where the convergence follows from Lemma \ref{lem:approxNi} and \eqref{eq:eq416}.
This proves  \eqref{eq:TN44} and hence as $N \to \infty$,
\begin{equation*}
\Tmctil^{N,4} \to \int_0^T \frac{1}{p(s)} \lambda_s \, ds \mbox{ in probability}.
\end{equation*}

Finally consider the last term $\Tmctil^{N,5}$.
Recalling $m_t$ defined in \eqref{eq:m_t}, we have
\begin{align*}
	\Tmctil^{N,5} & = \sum_{(i,j,k) \in \Smc_5} \int_0^T \frac{\xi_{ij}(s) \xi_{ik}(s)}{N_i^2(s)} \bbar_s(X^i_s,X^j_s) \cdot \bbar_s(X^i_s,X^k_s) \, ds \\
	& = \sum_{(i,j,k) \in \Smc_5} \int_0^T \frac{\xi_{ij}(s) \xi_{ik}(s)}{N_i^2(s)} \left( \bbar_s(X^i_s,X^j_s) \cdot \bbar_s(X^i_s,X^k_s) - m_s(X^j_s,X^k_s) \right) \, ds \\
	& \qquad + \sum_{(i,j,k) \in \Smc_5} \int_0^T \frac{\xi_{ij}(s) \xi_{ik}(s)}{N_i^2(s)} m_s(X^j_s,X^k_s) \, ds \\
	& \doteq \Tmctil^{N,5}_1 + \Tmctil^{N,5}_2.
\end{align*}
It follows from Lemma \ref{lem:Ttil} (see \eqref{eq:Ttilijk}) that $\EbfP |\Tmctil^{N,5}_1| \to 0$ as $N \to \infty$.
Let
\begin{equation*}
	\Tmchat^{N,5} \doteq \sum_{(i,j,k) \in \Smc_5} \int_0^T \frac{\xi_{ij}(s) \xi_{ik}(s)}{N^2 p_N^2(s)} m_s(X^j_s,X^k_s) \, ds.
\end{equation*}
We now show that as $N \to \infty$,
\begin{equation} \label{eq:TN5}
	\EbfP |\Tmctil^{N,5}_2 - \Tmchat^{N,5}|^2 \to 0.
\end{equation}
To see this, as before, it suffices to consider the summation over ordered indices $i < j < k$.
Note that
\begin{align*}
	& \EbfP \left( \sum_{1 \le i < j < k \le N} \int_0^T \xi_{ij}(s) \xi_{ik}(s) \left( \frac{1}{N_i^2(s)} - \frac{1}{N^2 p_N^2(s)} \right) m_s(X^j_s,X^k_s) \, ds   \right)^2 \\
	& \quad \le N \EbfP \sum_{i=1}^N \left( \sum_{1 \le j < k \le N} \int_0^T \xi_{ij}(s) \xi_{ik}(s) \left( \frac{1}{N_i^2(s)} - \frac{1}{N^2 p_N^2(s)} \right) m_s(X^j_s,X^k_s) \, ds   \right)^2 \\
	& \quad = N \EbfP \sum_{i=1}^N \sum_{1 \le j < k \le N} \left( \int_0^T \xi_{ij}(s) \xi_{ik}(s) \left( \frac{1}{N_i^2(s)} - \frac{1}{N^2 p_N^2(s)} \right) m_s(X^j_s,X^k_s) \, ds   \right)^2 \\
	& \quad \le \kappa_4 N^4 \EbfP \int_0^T \left( \frac{1}{N_1^2(s)} - \frac{1}{N^2 p_N^2(s)} \right)^2 ds \to 0,
\end{align*}
where the convergence follows from Lemma \ref{lem:approxNi}.
So \eqref{eq:TN5} holds.
Next split $\Tmchat^{N,5}$ as
\begin{align}
	\Tmchat^{N,5} & = \sum_{(i,j,k) \in \Smc_5, j<k} \int_0^T \frac{\xi_{ij}(s) \xi_{ik}(s) - p_N^2(s)}{N^2 p_N^2(s)} m_s(X^j_s,X^k_s) \, ds \notag \\
	& \qquad + \sum_{(i,j,k) \in \Smc_5, j>k} \int_0^T \frac{\xi_{ij}(s) \xi_{ik}(s) - p_N^2(s)}{N^2 p_N^2(s)} m_s(X^j_s,X^k_s) \, ds \notag \\
	& \qquad + \sum_{(i,j,k) \in \Smc_5} \frac{1}{N^2} \int_0^T m_s(X^j_s,X^k_s) \, ds \notag \\
	& \doteq \Tmchat^{N,5}_1 + \Tmchat^{N,5}_2 + \Tmchat^{N,5}_3. \label{eq:Tmchat_N_5_3}
\end{align}
It follows from Condition \ref{cond:cond2} that as $N \to \infty$
\begin{align*}
	\EbfP |\Tmchat^{N,5}_1|^2 & = \sum_{(i,j,k) \in \Smc_5, j<k} \EbfP \left( \int_0^T \frac{\xi_{ij}(s) \xi_{ik}(s) - p_N^2(s)}{N^2 p_N^2(s)} m_s(X^j_s,X^k_s) \, ds \right)^2 \\
	& \le \kappa_5 \sum_{(i,j,k) \in \Smc_5, j<k} \int_0^T \EbfP \left( \frac{\xi_{ij}(s) \xi_{ik}(s) - p_N^2(s)}{N^2 p_N^2(s)} \right)^2 \, ds \\
	& \le \frac{\kappa_6}{N \pbar_N^4} \to 0.
\end{align*}
Similarly $\EbfP |\Tmchat^{N,5}_2|^2 \to 0$ as $N \to \infty$.
Combining the above convergence results, we have
$$J^{N,2}(T) - \Tmchat^{N,5}_3 - \int_0^T \frac{\lambda_s}{p(s)}  \, ds \to 0$$
in probability as $N\to \infty$.  The result now follows on observing that
\begin{equation*}
	\Jtil^{N,2}(T) = \Tmchat^{N,5}_3 + \int_0^T \frac{\lambda_s}{p(s)}  \, ds. \qedhere
\end{equation*}
\end{proof}
From Lemma \ref{lem:JN1_JN1_tilde} and \ref{lem:tiljn2} we have that
\begin{equation}
	\label{eq:joint_new}
	\left(J^{N,1}(T), J^{N,2}(T)\right) = \left( \Jtil^{N,1}(T) + \Rmc^{N,1}, \Jtil^{N,2}(T) + \Rmc^{N,2} \right),
\end{equation}
where $\Rmc^{N,1}, \Rmc^{N,2} \to 0$ in probability as $N \to \infty$.






\subsubsection{Asymptotics of $(\etatil^N(\phi), \Jtil^{N,1}(T), \Jtil^{N,2}(T))$}
\label{sec:asymptotics_joint}

Define for $\zeta \in \Dmb([0,T]:\{0,1\})$ and $\nu\otimes \nu$ a.e.\ $(\omega,\omega') \in \Omega_d^2$,
\begin{align*}
	u_N(\zeta,\omega,\omega') & \doteq \int_0^T \frac{\zeta(s)-p_N(s)}{p_N(s)} \bbar_s(X_{*,s}(\omega),X_{*,s}(\omega')) \cdot d W_{*,s}(\omega) \\
	& \quad + \int_0^T \frac{\zeta(s)-p_N(s)}{p_N(s)} \bbar_s(X_{*,s}(\omega'),X_{*,s}(\omega)) \cdot d W_{*,s}(\omega').
\end{align*}
Let, with $\{V^i\}$ and $\{\xi_{ij}\}$ as in Section \ref{sec:canonical_processes},
\begin{equation}
	\label{eq:U_N}
	U_N \doteq \frac{1}{N} \sum_{1 \le i < j \le N} u_N(\xi_{ij},V^i,V^j).
\end{equation}
The following result is  key for studying the asymptotics of $(\etatil^N(\phi), \Jtil^{N,1}(T), \Jtil^{N,2}(T))$.
The special case where $\xi_{ij}(s) = \xi_{ij}(0)$ for all $s \in [0,T]$, $i,j \in \Nbd$ and $\phi_k=0$ for all $k\neq 2$ was considered by Janson in the study of incomplete $U$-statistics~\cite{Janson1984}.
%Proof of the case where $\{\xi_{ij}\}$ are time dependent is similar, however we provide the argument for completeness. 

Recall $Z$ and $\{I_k(\cdot)\}$ introduced in Section \ref{sec:complete_pf_CLT}.
\begin{lemma} \label{lem:key}
	Let $\{ \phi_k \}_{k=1}^\infty$ be such that $\phi_k \in L^2_{c,sym}(\nu^{\otimes k})$ for each $k \ge 1$. 
	Then the following convergence holds as $N \to \infty$:
	\begin{equation*}
		\left( U_N, \left( N^{-\frac{k}{2}} \Umc^N_k (\phi_k) \right)_{k \ge 1} \right) \Rightarrow \left( Z, \left( \frac{1}{k!} I_k(\phi_k) \right)_{k \ge 1} \right)
	\end{equation*}
	as a sequence of $\Rmb^\infty$-valued random variables, where $Z$ and $\{I_k(\cdot)\}_{k \ge 1}$ are as introduced in Section \ref{sec:complete_pf_CLT}, and $\Umc^N_k(\cdot)$ is as defined in Section \ref{sec:asymp_symmetric_statistics} with $Y_i$ replaced by $V_i$.
\end{lemma}





%Motivated by Janson's study on incomplete $U$-statistics \cite{Janson1984}, the following lemma is the key to studying the asymptotics of $\Jtil^{N,1}$.
%The following result in the case where $\xi_{ij}(s) = \xi_{ij}(0)$ for all $s \in [0,T]$ and $i,j \in \Nbd$ was obtained by Janson in the study of incomplete $U$-statistics~\cite{Janson1984}.
%Proof of the case where $\{\xi_{ij}\}$ are time dependent is similar, however we provide the argument for completeness. 
%This will be key to studying the asymptotics of $\Jtil^{N,1}$.

%\begin{lemma} \label{lem:key}
%	As $N \to \infty$,
%	\begin{equation*}
%		\sum_{1 \le i \ne j \le N} \int_0^T \frac{\xi_{ij}(s)}{N p_N(s)} \bbar_s(X^i_s,X^j_s) \cdot d W^i_s \Rightarrow Z + I_2(h^{sym}),
%	\end{equation*}
%	where $Z \sim \Nmc(0,\sigma^2)$ and $\{I_k(\cdot)\}_{k \ge 1}$ is as defined in Section \ref{sec:asymp_symmetric_statistics} with $\Smb = \Cmc_d$ and $\theta = \nu$.
%	Moreover, $Z$ is independent of $\{I_k(\cdot)\}_{k \ge 1}$.
%\end{lemma}

\begin{proof}
%	
%	We claim that for each $f \in L_{c,sym}^2(\nu^{\otimes 2})$, as $N \to \infty$
%	\begin{equation*} 
%%		\label{eq:keyincomplete}
%		\left(U_N,\Util_N(f)\right) \Rightarrow (Z,I_2(f)),
%	\end{equation*}
%	where $Z$ and $I_2(f)$ are as stated in part $(a)$.
%	Suppose for now that the claim holds.
%	Part $(a)$ follows directly.
%	For part $(b)$, fix $g \in L_c^2(\nu)$ and take $f = g \otimes g$.
%	From \eqref{eq:MWI_formula} it follows that $I_2(f) = [I_1(g)]^2 - \|g\|_{L_c^2(\nu)}^2$.
%	This together with the claim implies that $Z$ is independent of $[I_1(g)]^2$.
%	Part $(b)$ then follows on applying elementary properties of multivariate normal distribution.  
%	
%	Now we proceed to verify the above claim.
	The proof uses similar conditioning arguments as in Janson \cite{Janson1984} (see Lemma $2$ and Theorem $1$ therein).
	Fix $K \in \Nmb$, $\phi_k \in L^2_{c,sym}(\nu^{\otimes k})$ and $t, s_k \in \Rmb$ for $k=1,\dotsc,K$.
	Denote by $\EbfPV$ the conditional expectation under $\Pmb^N$ given $(V^i)_{i=1}^N$.	
	Since the collection $\{u_N(\xi_{ij},V^i,V^j)\}_{i<j}$ is conditionally independent given $(V^i)_{i=1}^N$, and $\EbfPV [u_N(\xi_{ij},V^i,V^j)] = 0$ for each $i<j$, we have
	\begin{equation*}
%		\tilde{U}_N & = \EbfP \left[U_N | V^1,\dotsc,V^N\right] = \frac{1}{N} \sum_{1 \le i < j \le N} \EbfP \left[u_N(\xi_{ij},V^i,V^j) | V^1,\dotsc,V^N\right], \\
		\sigma_N^2 \doteq \EbfPV [U_N^2] = \frac{1}{N^2} \sum_{1 \le i < j \le N} \EbfPV [u_N^2(\xi_{ij},V^i,V^j)].
	\end{equation*}
	Recall $\sigma^2$ defined in \eqref{eq:sigmasq}.
	It follows from \eqref{eq:eq416} that as $N \to \infty$,
	\begin{equation*}
		\EbfP [\sigma_N^2] = \frac{1}{N^2} \sum_{1 \le i < j \le N} \EbfP [u_N^2(\xi_{ij},V^i,V^j)] = \frac{2}{N^2} \sum_{1 \le i < j \le N} \int_0^T \frac{1-p_N(s)}{p_N(s)} \lambda_s \, ds \to \sigma^2
	\end{equation*}
	and
	\begin{align*}
		\EbfP \left(\sigma_N^2 - \EbfP [\sigma_N^2]\right)^2 & = \frac{1}{N^4} \sum_{1 \le i < j \le N} \EbfP \left( \EbfPV [u_N^2(\xi_{ij},V^i,V^j)] - \EbfP [u_N^2(\xi_{ij},V^i,V^j)] \right)^2 \\
		& \le \frac{N(N-1)}{2N^4} \EbfP [u_N^4(\xi_{12},V^1,V^2)] \to 0.
	\end{align*}
	So $\sigma_N^2 \to \sigma^2$ in probability.
	Suppose without loss of generality that $\sigma^2 > 0$, since otherwise we have that $Z = 0$, $U_N \to 0$ in probability as $N \to \infty$ and the desired convergence holds trivially by Theorem \ref{thm:Dynkin}.
	Also note that as $N \to \infty$,
	\begin{equation*}
		\EbfP \sum_{1 \le i < j \le N} \EbfPV \left|\frac{u_N(\xi_{ij},V^i,V^j)}{N}\right|^4 = \frac{N(N-1)}{2N^4} \EbfP [u_N^4(\xi_{12},V^1,V^2)] \to 0.
	\end{equation*}
	Hence the Lyapunov's condition for CLT (see~\cite{Billingsley1995probability}, Theorem 27.3) holds with $\delta=2$:
	\begin{equation*}
		\lim_{N \to \infty} \frac{1}{\sigma_N^{2+\delta}} \sum_{1 \le i < j \le N} \EbfPV \left|\frac{u_N(\xi_{ij},V^i,V^j)}{N}\right|^{2+\delta} = 0,
	\end{equation*}
	where the convergence is in probability.
	It then follows from standard proofs of CLT and a subsequence argument that for each $t \in \Rmb$,
	\begin{equation*}
		\EbfPV \left[e^{itU_N}\right] - e^{-\half t^2 \sigma_N^2} \to 0 
	\end{equation*}
	in probability as $N \to \infty$, which together with the convergence of $\sigma_N^2 \to \sigma^2$ implies that 
	\begin{equation}
		\EbfPV \left[ e^{itU_N} \right] \to e^{-t^2 \sigma^2/2}\label{eq:eq135}
	\end{equation}
	in probability as $N \to \infty$.
	Now let $(t, s_1, \ldots s_K) \mapsto \varphi_N(t,s_1,\dotsc,s_K)$ be the characteristic function of $$(U_N,N^{-\frac{1}{2}} \Umc^N_1 (\phi_1),\dotsc,N^{-\frac{K}{2}} \Umc^N_K (\phi_K)),$$ and $$\varphi(t,s_1,\dotsc,s_K) \doteq e^{-\half t^2\sigma^2} \psi(s_1,\dotsc,s_K),\;\; (t,s_1,\dotsc,s_K)\in \Rmb^{K+1}$$
	 be that of $(Z,I_1(\phi_1),\dotsc,\frac{1}{K!} I_K(\phi_K))$.
	It follows from Theorem \ref{thm:Dynkin} that for all $(s_1,\dotsc,s_K)\in \Rmb^{K}$
	\begin{equation}\EbfP [e^{i \sum_{k=1}^K s_k N^{-\frac{k}{2}} \Umc^N_k (\phi_k)}] \to \psi(s_1,\dotsc,s_K) \mbox{ as }
		N \to \infty.
		\label{eq:eq137}
	\end{equation}
	Thus as $N \to \infty$,
	\begin{align*}
		\varphi_N(t,s_1,\dotsc,s_K) - \varphi(t,s_1,\dotsc,s_K) & = \EbfP \left[ e^{itU_N + i \sum_{k=1}^K s_k N^{-\frac{k}{2}} \Umc^N_k (\phi_k)} - e^{-\half t^2\sigma^2} \psi(s_1,\dotsc,s_K) \right] \\
		& = \EbfP \left[ \left( \EbfPV \left[ e^{itU_N}\right] - e^{-\half t^2\sigma^2} \right) e^{i \sum_{k=1}^K s_k N^{-\frac{k}{2}} \Umc^N_k (\phi_k)} \right] \\
		& \quad + \left(\EbfP [e^{i \sum_{k=1}^K s_k N^{-\frac{k}{2}} \Umc^N_k (\phi_k)}] - \psi(s_1,\dotsc,s_K)\right) e^{-\half t^2\sigma^2} \\
		& \to 0,
	\end{align*}
	where the convergence follows from \eqref{eq:eq135} and \eqref{eq:eq137}.
	This completes the proof.	
\end{proof}
Now we  complete the proof of Proposition \ref{prop:key_joint_cvg}.
From \eqref{eq:JN1til}, \eqref{eq:U_N} and \eqref{eq:h_sym} we can write
\begin{align*} 
	\Jtil^{N,1}(T) &= \sum_{1 \le i < j \le N} \left( \int_0^T \frac{\xi_{ij}(s)-p_N(s)}{N p_N(s)} \bbar_s(X^i_s,X^j_s) \cdot d W^i_s + \int_0^T \frac{\xi_{ij}(s)-p_N(s)}{N p_N(s)} \bbar_s(X^j_s,X^i_s) \cdot d W^j_s \right) \\
	& \quad + \frac{1}{N} \sum_{1 \le i < j \le N} \left( \int_0^T \bbar_s(X^i_s,X^j_s) \cdot d W^i_s + \int_0^T \bbar_s(X^j_s,X^i_s) \cdot d W^j_s \right) \\
	& \quad + \sum_{i=1}^N \int_0^T \frac{1}{N p_N(s)} \bbar_s(X^i_s,X^i_s) \cdot d W^i_s \\
%	& = \frac{1}{N} \sum_{1 \le i < j \le N} u_N(\xi_{ij},V^i,V^j) + \frac{2}{N} \sum_{1 \le i < j \le N} h^{sym}(V^i,V^j) + \sum_{i=1}^N \int_0^T \frac{1}{N p_N(s)} \bbar_s(X^i_s,X^i_s) \cdot d W^i_s\\
	& = U_N + \frac{2}{N} \Umc^N_2(h^{sym}) + \sum_{i=1}^N \int_0^T \frac{1}{N p_N(s)} \bbar_s(X^i_s,X^i_s) \cdot d W^i_s.
\end{align*}
It follows from Condition \ref{cond:cond2} and law of large numbers that as $N \to \infty$
\begin{equation*}
	\sum_{i=1}^N \int_0^T \frac{1}{N p_N(s)} \bbar_s(X^i_s,X^i_s) \cdot d W^i_s \Rightarrow 0.
\end{equation*}
Also, from  \eqref{eq:eq114}  we have, with $l$ as in \eqref{eq:l},
\begin{equation*}
	\Jtil^{N,2}(T)-\int_0^T \frac{\lambda_s}{p(s)}  \, ds = \sum_{1 \le j \ne k \le N} \frac{N-2}{N^2} \int_0^T m_s(X^j_s,X^k_s) \, ds  = \frac{2(N-2)}{N^2} \Umc_2^N(l).
\end{equation*}
Combining above three displays with \eqref{eq:joint_new}, noting that
$\etatil^N(\phi) =  N^{-\frac{1}{2}} \Umc^N_1 (\phi)$,
and applying Lemma \ref{lem:key} gives us \eqref{eq:joint_cvg}.
This completes the proof of Proposition \ref{prop:key_joint_cvg}. \qed



%---------------------------------------------------------

\bigskip


\section*{Acknowledgement}
SB has been partially supported by NSF-DMS grants 1310002, 160683, 161307 and SES grant 1357622.
AB and RW have been partially supported by the National Science Foundation (DMS-1305120), the Army Research Office (W911NF-14-1-0331) and DARPA (W911NF-15-2-0122).


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\setcounter{equation}{0}
\appendix
\numberwithin{equation}{section}
% \section{Auxiliary Results}
% 
% 
% \section{Appendix} 
%\label{sec:appendix}

\section{Proof of Corollary \ref{cor:LLN}} \label{sec:pf_Cor_LLN}

(a) Fix $\alpha \in \Kbd$ and distinct $i_\alpha, i_\alpha' \in \Nalpha$.
For each fixed $g \in \Cmb_b(\Cmc_d)$ and $x \in \Cmc_d$, let $g^c(x) \doteq g(x) - \langle g, \mu^\alpha \rangle$.
Then as $N \to \infty$
\begin{align*}
	\Ebf \left( \langle g, \mu^{\alpha,N} \rangle - \langle g, \mu^\alpha \rangle \right)^2 & = \Ebf \left(\frac{1}{N_\alpha} \sum_{i \in \Nalpha} g^c(Z^{i,N})\right)^2 \\
	& = \frac{1}{N_\alpha^2} \sum_{i,j \in \Nalpha} \Ebf \left( g^c(Z^{i,N}) g^c(Z^{j,N}) \right) \\
	& = \frac{1}{N_\alpha} \Ebf \left( g^c(Z^{i_\alpha,N}) \right)^2 + \frac{N_\alpha-1}{N_\alpha} \Ebf \left( g^c(Z^{i_\alpha,N}) g^c(Z^{i_\alpha',N}) \right) \\
	& \le \frac{1}{\Nbar} + \Ebf \left( g^c(Z^{i_\alpha,N}) g^c(Z^{i_\alpha',N}) \right),
\end{align*}
which goes to $0$ as $N \to \infty$ by Corollary \ref{cor:poc} and Condition \ref{cond:cond1}. 
This proves part (a).

(b) 
Fix $\alpha, \gamma \in \Kbd$, $i \in \Nmb_\alpha$ and $i_\gamma \in \Nmb_\gamma$.
Abbreviate $\xi_{ij}^N(0), N_{i,\gamma}(0), p_{\alpha\gamma,N}(0)$ as  $\xi_{ij}, N_{i,\gamma}, p_{\alpha\gamma,N}$.
Let $\mubar^{i,\gamma,N} \doteq \frac{1}{N_{i,\gamma}} \sum_{j \in \Ngamma} \xi_{ij} \delta_{X^j} \one_{\{N_{i,\gamma}>0\}}$.
%Recall that $d_{BL}(\cdot,\cdot)$ is the bounded-Lipschitz metric on $\Pmc(\Cmc_d)$.
It suffices to show that $d_{BL}(\mu^{i,\gamma,N},\mubar^{i,\gamma,N}) \Rightarrow 0$ and $\mubar^{i,\gamma,N} \Rightarrow \mu^\gamma$ as $N \to \infty$.
%Recall that $\Bmb_1(\Cmc_d)$ is the space of bounded Lipschitz functions $f$ on $\Cmc_d$ such that $\|f\|_\infty \le 1$ and $\|f\|_L \le 1$, where $\|\cdot\|_L$ is the Lipschitz norm.
%For each $f \in \Bmb_1(\Cmc_d)$, let $f^c(x) = f(x) - \langle f, \mu \rangle$ for $x \in \Cmc_d$.
Note that
\begin{align*}
	\Ebf \, d_{BL}(\mu^{i,\gamma,N},\mubar^{i,\gamma,N}) & = \Ebf \sup_{\|g||_{BL} \le 1} \left| \langle g, \mu^{i,\gamma,N} \rangle - \langle g, \mubar^{i,\gamma,N} \rangle \right| \\
	& = \Ebf \sup_{\|g||_{BL} \le 1} \left| \sum_{j \in \Ngamma} \frac{\xi_{ij}}{N_{i,\gamma}} \one_{\{N_{i,\gamma}>0\}} \left( g(Z^{j,N}) - g(X^j) \right) \right| \\
	& \le \sum_{j \in \Ngamma} \Ebf \frac{\xi_{ij}}{N_{i,\gamma}} \one_{\{N_{i,\gamma}>0\}} \| Z^{j,N} - X^j \|_{*,T}.
%	& = \sum_{j \in \Ngamma} \Ebf \frac{\xi_{ji}}{N_j} \| Z^{i,\gamma,N} - X^i \|_{*,T} \\
%	& \le \Ebf \left( \sum_{j \in \Ngamma} \frac{\xi_{ij}}{N_j} - 1 \right) \| Z^{i,\gamma,N} - X^i \|_{*,T} + \Ebf \| Z^{i,\gamma,N} - X^i \|_{*,T} \\
%	& \le \sqrt{ \Ebf \left( \sum_{j \in \Ngamma} \frac{\xi_{ij}}{N_j} - 1 \right)^2 \Ebf \| Z^{i,\gamma,N} - X^i \|_{*,T}^2 } + \Ebf \| Z^{i,\gamma,N} - X^i \|_{*,T} \\
%	& \le \frac{2\sqrt{3}KT}{\sqrt{N\pbar_N}} + \frac{\kappa}{\sqrt{N\pbar_N}} \to 0
\end{align*}
By similar arguments as in \eqref{eq:TN2_gamma}--\eqref{eq:TN2_temp} we see that the above display can be bounded by
\begin{align*}
	& \left(\Ebf \left( \sum_{k \in \Nalpha} \frac{N_\gamma \xi_{k\igamma}}{N_\alpha N_{k,\gamma}} \one_{\{N_{k,\gamma} > 0\}} - 1 \right)^2 \Ebf \left\|Z^{\igamma,N} - X^\igamma\right\|^2_{*,T}\right)^{1/2} + \Ebf \left\|Z^{\igamma,N} - X^\igamma\right\|_{*,T} \\
	& \quad \le \kappa \left(\frac{1}{N_\alpha p_{\alpha\gamma,N}} + e^{-N_\gamma p_{\alpha\gamma,N}}\right)^{1/2} + \Ebf \left\|Z^{\igamma,N} - X^{\igamma}\right\|_{*,T} \\
	& \quad \le \kappa \left(\frac{1}{\Nbar \pbar_N} + e^{-\Nbar \pbar_N}\right)^{1/2} + \Ebf \left\|Z^{\igamma,N} - X^{\igamma}\right\|_{*,T},
\end{align*}
where the first inequality is from Lemma \ref{lem:prep_2} and \eqref{eq:ZiXi_bd}. From Theorem \ref{thm:NpN_rate} and Condition \ref{cond:cond1}, the last quantity
goes to $0$ as $N \to \infty$.
Thus
%\begin{equation} \label{eq:appendix of mu mu til}
$d_{BL}(\mu^{i,\gamma,N},\mubar^{i,\gamma,N}) \Rightarrow 0$ as $N \to \infty$.
%\end{equation}

Next we show that $\mubar^{i,\gamma,N} \Rightarrow \mu^\gamma$ as $N \to \infty$.
Recall that for $g \in \Cmb_b(\Cmc_d)$ and $x \in \Cmc_d$,  $g^c(x) \doteq g(x) - \langle g, \mu^\gamma \rangle$.
Then as $N \to \infty$,
\begin{align*}
	\Ebf \left( \langle g, \mubar^{i,\gamma,N} \rangle - \langle g, \mu^\gamma \rangle \one_{\{N_{i,\gamma}>0\}} \right)^2 & = \Ebf \left( \sum_{j \in \Ngamma} \frac{\xi_{ij}}{N_{i,\gamma}} \one_{\{N_{i,\gamma}>0\}} g^c(X^j) \right)^2 \\
	& = \Ebf \sum_{j \in \Ngamma} \frac{\xi_{ij}}{N_{i,\gamma}^2} \one_{\{N_{i,\gamma}>0\}} \left( g^c(X^j) \right)^2.
\end{align*}	
The expression on the right can be bounded above by	
\begin{align*} 4 \|g\|_\infty^2 \Ebf \sum_{j \in \Ngamma} \frac{\xi_{ij}}{N_{i,\gamma}^2} \one_{\{N_{i,\gamma}>0\}}  = 4 \|g\|_\infty^2 \Ebf \frac{1}{N_{i,\gamma}} \one_{\{N_{i,\gamma}>0\}}  \le 4 \|g\|_\infty^2 \Ebf \frac{2}{N_{i,\gamma}+1} \le \frac{8\|g\|_\infty^2}{N_\gamma p_{\alpha\gamma,N}}  \le \frac{8\|g\|_\infty^2}{\Nbar \pbar_N}  \to 0,
\end{align*}
where the second inequality follows from Lemma \ref{lem:prep_1}.
Also as $N \to \infty$,
\begin{equation*}
	\Ebf \left( \langle g, \mu^\gamma \rangle \one_{\{N_{i,\gamma}=0\}} \right)^2 \le \|g\|_\infty^2 (1-p_{\alpha\gamma,N})^{N_\gamma} \le \|g\|_\infty^2 e^{-N_\gamma p_{\alpha\gamma,N}} \le \|g\|_\infty^2 e^{-\Nbar\pbar_N} \to 0.
\end{equation*}
Combing above two convergence results implies that $\mubar^{i,\gamma,N} \Rightarrow \mu^\gamma$ as $N \to \infty$, and part (b) follows. \qed

\section{Proof of Lemmas from Section \ref{sec:pre_results}}
\label{sec:pf_section_5}

In this section we give the proofs of Lemmas \ref{lem:prep_1}, \ref{lem:prep_2}, \ref{lem:prep_4}.
\subsection{Proof of Lemma \ref{lem:prep_1}}

%\begin{proof}
	Note that
	\begin{gather*}
		\Ebf \frac{1}{X+1} = \sum_{k=0}^n \frac{1}{k+1} \binom{n}{k} p^k q^{n-k} = \frac{1}{(n+1)p} \sum_{k=0}^n \binom{n+1}{k+1} p^{k+1} q^{n-k} = \frac{1-q^{n+1}}{(n+1)p}.
	\end{gather*}
	Hence for each $m \in \Nmb$,
	\begin{equation*}
		\Ebf \frac{1}{X+m} = \Ebf \frac{1}{X+1} \frac{X+1}{X+m} \le \Ebf \frac{1}{X+1} \frac{n+1}{n+m} = \frac{1-q^{n+1}}{(n+1)p} \frac{n+1}{n+m} = \frac{1-q^{n+1}}{(n+m)p} \le \frac{1}{(n+m)p}.
	\end{equation*}
	Similarly,
	\begin{align*}
	\Ebf \frac{1}{(X+1)^m} & \le \Ebf \frac{m^m}{(X+1)(X+2)\dotsm(X+m)} = \sum_{k=0}^n \frac{m^m}{(k+1)(k+2)\dotsm(k+m)} \binom{n}{k} p^k q^{n-k} \\
	& \le \frac{m^m}{(n+1)(n+2)\dotsm(n+m)p^m} \le \frac{m^m}{(n+1)^m p^m},
	\end{align*}
	which completes the proof.
	\qed
%\end{proof}

\subsection{Proof of Lemma \ref{lem:prep_2}}

%\begin{proof}
%	We first prove the case $\alpha \ne \gamma$.
	Fix $\alpha \ne \gamma$, $i_\gamma \in \Ngamma$ and $i_\alpha \in \Nalpha$.
	Since 
	\begin{equation*}
%		\label{eq:pf_prep3_exchange}
		\Lmc(\zeta_{ki_\gamma}, N_{k,\gamma}) = \Lmc(\zeta_{i_\alpha i_\gamma}, N_{i_\alpha,\gamma}) = \Lmc(\zeta_{i_\alpha j}, N_{i_\alpha,\gamma}), \quad k \in \Nalpha, j \in \Ngamma,
	\end{equation*} 
	we have 
	\begin{align*}
		\Ebf \sum_{k \in \Nalpha} \frac{N_\gamma \zeta_{ki_\gamma}}{N_\alpha N_{k,\gamma}} \one_{\{N_{k,\gamma} > 0\}} = \Ebf \frac{N_\gamma \zeta_{i_\alpha i_\gamma}}{N_{i_\alpha,\gamma}} \one_{\{N_{i_\alpha,\gamma} > 0\}} = \Ebf \sum_{k \in \Ngamma} \frac{\zeta_{i_\alpha k}}{N_{i_\alpha,\gamma}} \one_{\{N_{i_\alpha,\gamma} > 0\}} = \Pbd(N_{i_\alpha,\gamma} > 0).
	\end{align*}
	Using this we can write
	\begin{align}
		& \Ebf \left( \sum_{k \in \Nalpha} \frac{N_\gamma \zeta_{ki_\gamma}}{N_\alpha N_{k,\gamma}} \one_{\{N_{k,\gamma} > 0\}} - 1 \right)^2 \notag \\
		% & = \Ebf \left( \sum_{k \in \Nalpha} \frac{N_\gamma \zeta_{ki_\gamma}}{N_\alpha N_{k,\gamma}} \one_{\{N_{k,\gamma} > 0\}} \right)^2 - 2 \Ebf \sum_{k \in \Nalpha} \frac{N_\gamma \zeta_{ki_\gamma}}{N_\alpha N_{k,\gamma}} \one_{\{N_{k,\gamma} > 0\}} + 1 \notag \\
		& = \frac{N_\gamma^2}{N_\alpha^2} \Ebf \sum_{k,l \in \Nalpha} \frac{\zeta_{ki_\gamma} \zeta_{li_\gamma}}{N_{k,\gamma} N_{l,\gamma}} \one_{\{N_{k,\gamma} > 0\}} \one_{\{N_{l,\gamma} > 0\}} - 2 \Pbd(N_{i_\alpha,\gamma} > 0) + 1 \notag \\
		& = \frac{N_\gamma^2}{N_\alpha^2} \Ebf \sum_{k \in \Nalpha} \frac{\zeta_{ki_\gamma}}{N_{k,\gamma}^2} \one_{\{N_{k,\gamma} > 0\}} + \frac{N_\gamma^2}{N_\alpha^2} \Ebf \sum_{k,l \in \Nalpha, k \ne l} \frac{\zeta_{ki_\gamma} \zeta_{li_\gamma}}{N_{k,\gamma} N_{l,\gamma}} \one_{\{N_{k,\gamma} > 0\}} \one_{\{N_{l,\gamma} > 0\}} + 2 \Pbd(N_{i_\alpha,\gamma} = 0) - 1 \notag \\
		& \doteq \sum_{n=1}^3 \Tmc^{N,n}_\gamma - 1, \label{eq:pf_prep3_key}
	\end{align}
	For $\Tmc^{N,1}_\gamma$, we have  by a straightforward conditioning argument,
	\begin{align}
		\Tmc^{N,1}_\gamma & \doteq \frac{N_\gamma^2}{N_\alpha^2} \sum_{k \in \Nalpha} \Ebf \frac{\zeta_{ki_\gamma}}{N_{k,\gamma}^2} \one_{\{N_{k,\gamma} > 0\}} = \frac{N_\gamma^2}{N_\alpha^2} \sum_{k \in \Nalpha} p_{\alpha\gamma,N} \Ebf \frac{1}{(N_{k,\gamma}-\zeta_{ki_\gamma}+1)^2} \notag \\
		& \le \frac{N_\gamma^2}{N_\alpha^2} N_\alpha p_{\alpha\gamma,N} \frac{4}{(N_\gamma p_{\alpha\gamma,N})^2} = \frac{4}{N_\alpha p_{\alpha\gamma,N}}, \label{eq:pf_prep3_bound1}
	\end{align}
	where the inequality follows from Lemma \ref{lem:prep_1}.
	For $\Tmc^{N,2}_\gamma$, using the independence of $(\zeta_{ki_\gamma}, N_{k,\gamma})$ and $(\zeta_{li_\gamma}, N_{l,\gamma})$ for different $k,l \in \Nalpha$, we have
	\begin{align}
		\Tmc^{N,2}_\gamma 
		& = \frac{N_\gamma^2}{N_\alpha^2} \sum_{k,l \in \Nalpha, k \ne l} \Ebf \left( \frac{\zeta_{ki_\gamma}}{N_{k,\gamma}} \one_{\{N_{k,\gamma} > 0\}} \right) \Ebf \left( \frac{\zeta_{li_\gamma}}{N_{l,\gamma}} \one_{\{N_{l,\gamma} > 0\}} \right) \notag \\
		& = \frac{N_\gamma^2}{N_\alpha^2} \sum_{k,l \in \Nalpha, k \ne l} p_{\alpha\gamma,N}^2 \Ebf \left( \frac{1}{N_{k,\gamma}-\zeta_{ki_\gamma}+1} \right) \Ebf \left( \frac{1}{N_{l,\gamma}-\zeta_{li_\gamma}+1} \right) \notag \\
		& \le \frac{N_\gamma^2}{N_\alpha^2} N_\alpha (N_\alpha-1) p_{\alpha\gamma,N}^2 \left( \frac{1}{N_\gamma p_{\alpha\gamma,N}} \right)^2 \le 1, \label{eq:pf_prep3_bound2}
	\end{align}
	where the first inequality once more follows from Lemma \ref{lem:prep_1}.
	Finally, for $\Tmc^{N,3}_\gamma$, we have
	\begin{equation}
		\label{eq:pf_prep3_bound3}
		\Tmc^{N,3}_\gamma \doteq 2 \Pbd(N_{i_\alpha,\gamma} = 0) = 2 (1 - p_{\alpha\gamma,N})^{N_\gamma} \le 2 e^{-N_\gamma p_{\alpha\gamma,N}}.
	\end{equation}		
	Plugging \eqref{eq:pf_prep3_bound1}--\eqref{eq:pf_prep3_bound3} into \eqref{eq:pf_prep3_key} gives the first statement when $\alpha \neq \gamma$. The case $\alpha =\gamma$ is immediate
	from the second statement which we prove next.

	Fix $i_\alpha \in \Nalpha$.
	Since
	\begin{equation}
		\label{eq:pf_prep3_exchange_new}
		\Lmc(\zeta_{ki_\alpha}, N_{k,\alpha}) = \Lmc(\zeta_{i_\alpha k}, N_{i_\alpha,\alpha}), \quad k \in \Nalpha,
	\end{equation} 
	we have
	\begin{equation*}
		\Ebf \sum_{k \in \Nalpha} \frac{\zeta_{ki_\alpha}}{N_{k,\alpha}} = \Ebf \sum_{k \in \Nalpha} \frac{\zeta_{i_\alpha k}}{N_{i_\alpha,\alpha}} = 1.
	\end{equation*}
	Using this we can write
	\begin{align}
		\Ebf \left( \sum_{k \in \Nalpha} \frac{\zeta_{ki_\alpha}}{N_{k,\alpha}} - 1 \right)^2 & = \Ebf \left( \sum_{k \in \Nalpha} \frac{\zeta_{ki_\alpha}}{N_{k,\alpha}} \right)^2 - 2 \Ebf \sum_{k \in \Nalpha} \frac{\zeta_{ki_\alpha}}{N_{k,\alpha}} + 1 = \Ebf \sum_{k,l \in \Nalpha} \frac{\zeta_{ki_\alpha} \zeta_{li_\alpha}}{N_{k,\alpha} N_{l,\alpha}} - 1 \notag \\
		& = \sum_{n=1}^4 \Ebf \sum_{(k,l) \in \Smc^{N,n}_\alpha} \frac{\zeta_{ki_\alpha} \zeta_{li_\alpha}}{N_{k,\alpha} N_{l,\alpha}} - 1 \doteq \sum_{n=1}^4 \Tmc^{N,n}_\alpha - 1, \label{eq:pf_prep3_key_new}
	\end{align}
	where $\Smc^{N,1}_\alpha$, $\Smc^{N,2}_\alpha$, $\Smc^{N,3}_\alpha$ and $\Smc^{N,4}_\alpha$ are collections of $(k,l) \in \Nalpha \times \Nalpha$ such that $\{ k = l \}$, $\{k \ne l, k=i_\alpha \}$, $\{ k \ne l, l=i_\alpha \}$ and $\{ i_\alpha,k,l \text{ distinct} \}$, respectively.
	For $\Tmc^{N,1}_\alpha$, it follows from \eqref{eq:pf_prep3_exchange_new} that
	\begin{equation} \label{eq:pf_prep3_bound1_new}
		\Tmc^{N,1}_\alpha \doteq \sum_{k \in \Nalpha} \Ebf \frac{\zeta_{ki_\alpha}}{N_{k,\alpha}^2} = \sum_{k \in \Nalpha} \Ebf \frac{\zeta_{i_\alpha k}}{N_{i_\alpha,\alpha}^2} = \Ebf \frac{1}{N_{i_\alpha,\alpha}} \le \frac{1}{N_\alpha p_{\alpha\alpha,N}},
	\end{equation}
	where the inequality follows from Lemma \ref{lem:prep_1}.
	For $\Tmc^{N,2}_\alpha$, using independence of $N_{i_\alpha,\alpha} - \zeta_{li_\alpha}$ and $N_{l,\alpha} - \zeta_{li_\alpha}$ for different $i_\alpha, l \in \Nalpha$, we have
	\begin{align}
		\Tmc^{N,2}_\alpha & = \sum_{l \in \Nalpha, l \ne i_\alpha} \Ebf \frac{\zeta_{li_\alpha}}{N_{i_\alpha,\alpha} N_{l,\alpha}} = \sum_{l \in \Nalpha, l \ne i_\alpha} p_{\alpha\alpha,N} \Ebf \frac{1}{(N_{i_\alpha,\alpha} + 1 - \zeta_{li_\alpha})(N_{l,\alpha} + 1 - \zeta_{li_\alpha})} \notag \\
		& = \sum_{l \in \Nalpha, l \ne i_\alpha} p_{\alpha\alpha,N} \Ebf \frac{1}{N_{i_\alpha,\alpha} + 1 - \zeta_{li_\alpha}} \Ebf \frac{1}{N_{l,\alpha} + 1 - \zeta_{li_\alpha}} \notag \\
		& \le (N_\alpha - 1) p_{\alpha\alpha,N} \frac{1}{(N_\alpha p_{\alpha\alpha,N})^2} \le \frac{1}{N_\alpha p_{\alpha\alpha,N}}, \label{eq:pf_prep3_bound2_new}
	\end{align}
	where the first inequality again follows from Lemma \ref{lem:prep_1}.
	Similarly for $\Tmc^{N,3}_\alpha$, we have
	\begin{equation} 
		\label{eq:pf_prep3_bound3_new}
		\Tmc^{N,3}_\alpha = \sum_{k \in \Nalpha, k \ne i_\alpha} \Ebf \frac{\zeta_{ki_\alpha}}{N_{i_\alpha,\alpha} N_{k,\alpha}} \le \frac{1}{N_\alpha p_{\alpha\alpha,N}}.
	\end{equation}
	Finally for $\Tmc^{N,4}_\alpha$, using the independence of $N_{k,\alpha}-\zeta_{ki_\alpha}-\zeta_{kl}$ and $N_{l,\alpha}-\zeta_{li_\alpha}-\zeta_{kl}$ for distinct $i_\alpha,k,l \in \Nalpha$, we have
	\begin{align}
%		\Tmc^{N,4}_\alpha & = \sum_{\substacl{1 \le k,l \le N \\ i,k,l \text{ distinct}}} \Ebf \frac{\zeta_{ik} \zeta_{il}}{N_k N_l} = \sum_{\substacl{1 \le k,l \le N \\ i,k,l \text{ distinct}}} p_N^2 \Ebf \frac{1}{(2+\Ntil_{k,-ki})(2+\Ntil_{l,-li})} \notag \\
		\Tmc^{N,4}_\alpha & = \sum_{(k,l) \in \Smc^{N,4}_\alpha} \Ebf \frac{\zeta_{ki_\alpha} \zeta_{li_\alpha}}{N_{k,\alpha} N_{l,\alpha}} = \sum_{(k,l) \in \Smc^{N,4}_\alpha} p_{\alpha\alpha,N}^2 \Ebf \frac{1}{(N_{k,\alpha}+1-\zeta_{ki_\alpha})(N_{l,\alpha}+1-\zeta_{li_\alpha})} \notag \\
		& \le \sum_{(k,l) \in \Smc^{N,4}_\alpha} p_{\alpha\alpha,N}^2 \Ebf \frac{1}{(N_{k,\alpha}+1-\zeta_{ki_\alpha}-\zeta_{kl})(N_{l,\alpha}+1-\zeta_{li_\alpha}-\zeta_{kl})} \notag \\
		& = \sum_{(k,l) \in \Smc^{N,4}_\alpha} p_{\alpha\alpha,N}^2 \Ebf \frac{1}{N_{k,\alpha}+1-\zeta_{ki_\alpha}-\zeta_{kl}} \Ebf \frac{1}{N_{l,\alpha}+1-\zeta_{li_\alpha}-\zeta_{kl}} \notag \\
		& \le (N_\alpha-1)(N_\alpha-2) p_{\alpha\alpha,N}^2 \left( \frac{1}{(N_\alpha-1) p_{\alpha\alpha,N}} \right)^2 \le 1, \label{eq:pf_prep3_bound4_new}
	\end{align}
	where the second inequality once more follows from Lemma \ref{lem:prep_1}.
	Plugging \eqref{eq:pf_prep3_bound1_new}--\eqref{eq:pf_prep3_bound4_new} into \eqref{eq:pf_prep3_key_new} completes the proof.
	\qed

\subsection{Proof of Lemma \ref{lem:prep_4}}

%\begin{proof}
	First note that the result holds trivially when $p_N = 0$ or $p_N = 1$.
	Now consider the case $p_N \in (0,1)$.
	For $t \ge 0$, it follows from Hoeffding's inequality that
	\begin{equation*}
		\Pbd \left( |Y - Np_N| > t + 1 \right) \le \Pbd \left( \left| \sum_{i=2}^N (\zeta_i - p_N) \right| > t \right) \le 2 e^{- \frac{2t^2}{N-1}}.
	\end{equation*}
	Taking $t = C_N(k)$ completes the proof.
	\qed
%\end{proof}

%We know that
%\begin{align*}
%\Ebf \left( \sum_{j=1}^N (\frac{\xi_{ij}}{N_j} - \frac{1}{N}) \right)^2 & = \sum_{j=1}^N \Ebf (\frac{\xi_{ij}}{N_j} - \frac{1}{N})^2 + \sum_{1 \le j \ne k \le N} \Ebf (\frac{\xi_{ij}}{N_j} - \frac{1}{N}) (\frac{\xi_{ik}}{N_k} - \frac{1}{N}) \\
%& = \Ebf \frac{1}{N_j} - \frac{1}{N} + N(N-1) \left( \Ebf \frac{\xi_{ij}\xi_{ik}}{N_j N_k} - \frac{1}{N^2} \right).
%\end{align*}
%For the first term, we have
%\begin{align}
%\Ebf \frac{1}{N_j} - \frac{1}{N p_N} & = \Ebf \frac{N p_N - N_j}{N_j N p_N} \le \Ebf \frac{|N_j - N p_N|}{N_j N p_N} \le \Ebf \frac{|N_j - N p_N|}{N p_N} \le \frac{\sqrt{Var(N_j)}}{N p_N} = \frac{\sqrt{N p_N q_N}}{N p_N} \le \frac{1}{\sqrt{N p_N}}
%\end{align}
%As for the second term, let $\Ntil_{i,-i} = \sum_{j \ne i} \xi_{ij}$, we have
%\begin{align*}
%\Ebf \frac{\xi_{ij}\xi_{ik}}{N_j N_k} 
%& = \Pmb(\xi_{jk} = 0) \Ebf \frac{\xi_{ij}\xi_{ik}}{\Ntil_j \Ntil_k} + \Pmb(\xi_{jk} = 1) \Ebf \frac{\xi_{ij}\xi_{ik}}{(1 + \Ntil_j)(1 + \Ntil_k)} \\
%& = q_N (\Ebf \frac{\xi_{ij}}{\Ntil_j})^2 + p_N (\Ebf \frac{\xi_{ij}}{1 + \Ntil_j})^2 
%\le (\Ebf \frac{\xi_{ij}}{\Ntil_j})^2 \\
%& = ( p_N \Ebf \frac{1}{1+ \text{Bernoulli}(N-2,p_N)})^2 
%= ( p_N \frac{1 - q_N^{N-1}}{(N-1)p_N})^2 \\
%& \le \frac{1}{(N-1)^2}.
%\end{align*}
%Combining above two inequalities gives us
%\begin{align*}
%\Ebf \left( \sum_{j=1}^N (\frac{\xi_{ij}}{N_j} - \frac{1}{N}) \right)^2 & = \Ebf \frac{1}{N_j} - \frac{1}{N} + N(N-1) \left( \Ebf \frac{\xi_{ij}\xi_{ik}}{N_j N_k} - \frac{1}{N^2} \right) \\
%& \le \frac{1}{\sqrt{N p_N}} + \frac{1}{N p_N} - \frac{1}{N} + N(N-1)(\frac{1}{(N-1)^2} - \frac{1}{N^2}) \\
%& = \frac{1}{\sqrt{N p_N}} + \frac{1}{N p_N} + \frac{1}{N-1}.
%\end{align*}


%By computing the second moment and using the fact that $\xi_{ij}$ and $X^i_s$ are independent, we can show $B$ and $C$ are both in the order of $\frac{1}{\sqrt{N}}$. As for $A$, since $(X^{i,N}, X^i)_{i \ge 1}$ are exchangeable pairs, we can have
%\begin{equation}\label{A}
%A \le K\Ebf \frac{1}{N} \sum_{j=1}^N |\xi_{ij}| \left(|X^{i,N}_s - X^i_s| + |X^{j,N}_s - X^j_s|\right) \le 2K \Ebf |X^{i,N}_t - X^i_t|_{*,s}.
%\end{equation}
%So we can get
%\begin{equation*}
%\Ebf |X^{i,N}_t - X^i_t|_{*,T} \le \int_0^T 2K \Ebf |X^{i,N}_t - X^i_t|_{*,s} \, ds + \frac{\kappa}{\sqrt{N}}.
%\end{equation*}


\section{A lemma on integral operators} \label{sec:restating}


Let $\Smb$ be a Polish space and $\nu \in \Pmc(\Smb)$.
Let $a(\cdot,\cdot) \in L^2(\nu \otimes \nu)$ and denote by $A$ the integral operator on $L^2(\nu)$ associated with $a$: $A \phi(x) \doteq \int_\Smb a(x,y) \phi(y) \, \nu(dy)$ for $x \in \Smb$ and $\phi \in L^2(\nu)$.
Then $A$ is a Hilbert-Schmidt operator.
Also, $AA^*$, and for $n \ge 2$, $A^n$, are trace class operators.
The following lemma is taken from  Shiga-Tanaka \cite{ShigaTanaka1985}.

\begin{lemma} \label{lem:Shiga_Tanaka}
	Suppose that \textnormal{Trace}$(A^n)=0$ for all $n \ge 2$.
	Then $\Ebf [e^{\frac{1}{2} I_2(f)}] = e^{\frac{1}{2} \textnormal{Trace} (AA^*)}$, where $f(x,y) \doteq a(x,y) + a(y,x) - \int_\Smb a(x,z) a(y,z) \, \nu(dz)$, and $I_2(\cdot)$ is the MWI defined as in Section \ref{sec:asymp_symmetric_statistics}.
	Moreover, $I-A$ is invertible and for any $\phi \in L^2(\nu)$, 
	$$\Ebf \left[\exp(iI_1(\phi) + \frac{1}{2} I_2(f))\right] = \exp\left\{-\frac{1}{2} (\| (I-A)^{-1} \phi \|^2_{L^2(\nu)} - \textnormal{Trace} (AA^*))\right\},$$
	 where $I$ is the identity operator on $L^2(\nu)$.
\end{lemma}




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% {\sc
% \bigskip
% \noi
% S. Bhamidi (Email: bhamidi@email.unc.edu)\\
% A. Budhiraja (Email: budhiraj@email.unc.edu)\\
% R. Wu (Email: wuruoyu@live.unc.edu)\\
% Department of Statistics and Operations Research\\
% University of North Carolina\\
% Chapel Hill, NC 27599, USA\\
% %
% %\skp
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% %\noi
% %
% %Department of Statistics and Operations Research\\
% %University of North Carolina\\
% %Chapel Hill, NC 27599, USA\\
% %
% %\skp
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% %\noi
% %
% %Department of Statistics and Operations Research\\
% %University of North Carolina\\
% %Chapel Hill, NC 27599, USA\\
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% }
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