Moment convergence in renewal theory
Alexander Iksanov\footnote{ Faculty of Cybernetics
01033 Kiev
Ukraine
and Matthias Meiners\footnote{ Institut f\"{u}r Mathematische Statistik
Westf\"{a}lische Wilhelms-Universit\"{a}t M\"{u}nster
Let $\xi_1, \xi_2, \ldots$ be independent copies of a positive random variable $\xi$, and let $S_k := \xi_1+\ldots+\xi_k$, $k \in \N_0$. Define $N(t) := \#\{k \in \N_0: S_k \leq t\}$. $(N(t))_{t \geq 0}$ is a renewal counting process. It is known that if $\xi$ is in the domain of attraction of a stable law of index $\alpha \in (1,2]$, then $N(t)$, suitably shifted and scaled, converges in distribution as $t \to \infty$ to a random variable with a stable law. We show that in this situation, also the first absolute moments converge to the first absolute moment of the limiting random variable. Further, the corresponding result for subordinators is established.
1208.3964.tex
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\title{Moment convergence in renewal theory}
\date{\today}

\author{Alexander Iksanov\footnote{ Faculty of Cybernetics, National T.\
Shevchenko University of Kiev, 01033 Kiev, Ukraine, e-mail:
iksan@univ.kiev.ua\newline A part of this work was carried out while A.
Iksanov was visiting M\"{u}nster in May and June 2012.
Grateful acknowledgment is made for financial support and
hospitality.},\ 
Alexander Marynych\footnote{Eindhoven University of Technology, The Netherlands, e-mail: O.Marynych@tue.nl}	\\
and Matthias Meiners\footnote{ Institut f\"{u}r Mathematische Statistik,
Westf\"{a}lische Wilhelms-Universit\"{a}t M\"{u}nster, 48149
M\"{u}nster, Germany, e-mail: mmeiners@math.uni-muenster.de.
Research supported by DFG SFB 878 ``Geometry, Groups and Actions''}}

\begin{document}

\thispagestyle{empty}
\maketitle

\begin{abstract}
Let $\xi_1, \xi_2, \ldots$ be independent copies of a positive random variable $\xi$, and let $S_k := \xi_1+\ldots+\xi_k$, $k \in \N_0$.
Define $N(t) := \#\{k \in \N_0: S_k \leq t\}$. $(N(t))_{t \geq 0}$ is a renewal counting process.
It is known that if $\xi$ is in the domain of attraction of a stable law of index $\alpha \in (1,2]$,
then $N(t)$, suitably shifted and scaled, converges in distribution as $t \to \infty$ to a random variable with a stable law.
We show that in this situation,
also the first absolute moments converge to the first absolute moment of the limiting random variable.
Further, the corresponding result for subordinators is established.
\end{abstract}


\section{Introduction and results}

Let $\xi_1, \xi_2, \ldots$ be i.i.d.\ copies of a
positive random variable $\xi$. Define
\begin{equation*}
S_0	:=	0,	\quad	S_k :=	\xi_1+\ldots+\xi_k, \quad	k \in \N,
\end{equation*}
and
\begin{equation*}
N(t)	~:=~	\#\{k\in\N_0: S_k \leq t\}	~=~	\inf\{k\in\N: S_k > t\},	\quad	t \geq 0.
\end{equation*}
The stochastic process $(N(t))_{t \geq 0}$ is called \emph{renewal counting process} associated with $(S_k)_{k \geq 0}$.

By $(S(t))_{t \geq 0}$ we denote a subordinator
(\textit{i.e.}\ an increasing L{\'e}vy process) with $S(0)=0$, zero drift, no killing and a L{\'e}vy measure $\nu$ that is concentrated on $\R_+$
(compound Poisson processes are not excluded).
Put
\begin{equation*}
T(s) ~:=~ \inf\{t \geq 0: S(t)>s\}, \quad	s \geq 0.
\end{equation*}
The stochastic process $(T(s))_{s \geq 0}$ is called \emph{first-passage time process} associated with $(S(t))_{t \geq 0}$.

It is known that under certain assumptions (precise references will be given below) which include
$\mu := \E \xi < \infty$ and ${\tt m}:=\E S(1)<\infty$ that
\begin{equation}	\label{eq:convergence of N(s)}
\frac{N(s)-\mu^{-1}s}{g_1(s)} ~\stackrel{\mathrm{d}}{\to}~ W_1, \quad s \to \infty
\end{equation}
and
\begin{equation}	\label{eq:convergence of T(s)}
\frac{T(s)-{\tt m}^{-1}s}{g_2(s)} ~\stackrel{\mathrm{d}}{\to}~ W_2, \quad s\to\infty
\end{equation}
for some random variables $W_j$ with $\E |W_j| < \infty$ for $j=1,2$ and suitable scaling functions $g_1, g_2$.
The purpose of this note is to check whether there is convergence of moments in these relations, for instance,
\begin{equation*}
\lim_{s \to \infty} \frac{\E |N(s)-\mu^{-1}s|}{g_1(s)}	~=~	\E |W_1|?
\end{equation*}
We do not claim the results stated below are new.
However, we have been unable to locate them in the literature.
Theorem 1.1 is an important ingredient in our forthcoming paper on the weak
convergence of shot noise processes \cite{IMM}.
Theorem 1.2 is essentially used in the proof of Theorem 3.3 in \cite{GneIks}.

\begin{Theorem}	\label{Thm:moment convergence}
Let $\mu:=\E \xi < \infty$. Then the following assertions hold:
\begin{itemize}
	\item[(A1)]
		If $\sigma^2:= \Var \xi<\infty$, then
		\begin{equation*}
		\lim_{s \to \infty} \frac{\E |N(s)-\mu^{-1}s|}{\sqrt{s}}
		~=~ \frac{\sigma}{\mu^{3/2}}\E |W| ~=~ \sigma \sqrt{2 \over \pi \mu^3}
		\end{equation*}
		where $W$ is a random variable with standard normal law.
	\item[(A2)]
		Suppose $\sigma^2=\infty$ and
		\begin{equation*}
		\int_{[0,\,x]}y^2\Prob\{\xi \in \dy\}	~\sim~	\ell(x),	\quad x \to \infty,
		\end{equation*}
		for some $\ell$ slowly varying at $\infty$. Then
		\begin{equation*}
		\lim_{s \to \infty} \frac{\E |N(s)-\mu^{-1}s|}{c(s)} ~=~ \frac{1}{\mu^{3/2}} \E |W| ~=~ \sqrt{\frac{2}{\pi\mu^3}}
		\end{equation*}
		where $c(x)$ is a positive function satisfying $\lim_{x \to \infty} x \ell(c(x))/c(x)^2=1$,
		and $W$ is a random variable with standard normal law.
	\item[(A3)]
		Suppose $\Prob\{\xi>x\} \sim x^{-\alpha}\ell(x)$ as $x \to \infty$ for some $\alpha \in (1,2)$ and some $\ell$ slowly varying at $\infty$.
		Then
		\begin{equation*}
		\lim_{s \to \infty} \frac{\E |N(s)-\mu^{-1}s|}{c(s)}
		~=~	\frac{\E |W|}{\mu^{1+1/\alpha}}
		~=~	\frac{2\Gamma(1-\frac{1}{\alpha})\Gamma(1-\alpha)^{1/\alpha} \sin(\frac{\pi}{\alpha})}{\pi\mu^{1+1/\alpha}}
		\end{equation*}
		where $c(x)$ is a positive function such that $\lim_{x \to \infty} x \ell(c(x)) c^{-\alpha}(x) = 1$,
		and $W$ is a random variable with characteristic function
		\begin{equation}	\label{eq:characteristic function}
		t \mapsto \exp\big\{-|t|^\alpha
		\Gamma(1-\alpha)(\cos(\pi\alpha/2)+ \i \sin(\pi\alpha/2)\, {\rm sgn}(t))\big\},	\quad t \in \R
		\end{equation}
		where $\Gamma(\cdot)$ denotes Euler's gamma function.
\end{itemize}
\end{Theorem}

Assertion (A1) is stated for completeness only.
The result is well known and can be found, for instance, in \cite[Theorem 3.8.4(i)]{Gut}.
The proof of assertion (A1) is included in this note, for it requires no extra work in the given framework.

Knowing already the weak convergence relation \eqref{eq:convergence of N(s)},
uniform integrability of the family $|N(s)-s/\mu|/c(s)$, $s \geq 1$ would suffice to conclude convergence of the first absolute moments.
This approach has been used in \cite{Gut}. However, the relevant results on uniform integrability presented in the cited book
(such as Theorem 3.7.3 there) are not strong enough for our purposes.

\begin{Theorem}	\label{Thm:moment convergence subordinator}
Assume ${\tt m}:=\E S(1)<\infty$. Then the following assertions hold:
\begin{itemize}
	\item[(B1)]
		If ${\tt b}^2= \Var S(1)=\int_{[0,\,\infty)} x^2 \, \nu(\dx)<\infty$, then
		\begin{equation*}
		\lim_{s \to \infty} \frac{\E |T(s)-{\tt m}^{-1}s|}{\sqrt{s}} ~=~ \frac{{\tt b}}{{\tt m}^{3/2}} \E |W| ~=~ {\tt b} \sqrt{\frac{2}{\pi{\tt m}^3}}
		\end{equation*}
		where $W$ is a random variable with standard normal law.
	\item[(B2)]
		Suppose
		\begin{equation}	\label{2}
		{\tt b}^2=\infty	\quad \text{and} \quad \int_{[0,\,x]} y^2 \, \nu(\dy)
		~\sim~ \ell(x), \quad x \to \infty,
		\end{equation}
		for some $\ell$ slowly varying at $\infty$. Then
		\begin{equation*}
		\lim_{s \to \infty} \frac{\E |T(s)-{\tt m}^{-1}s|}{c(s)}	~=~	\frac{1}{{\tt m}^{3/2}}\E |W|	~=~	\sqrt{{2}{\pi{\tt m}^3}}
		\end{equation*}
		where $c(t)$ is a positive function such that $\lim_{x \to \infty} x \ell(c(x))c^{-2}(x) = 1$,
		and $W$ is a random variable with standard normal law.
	\item[(B3)]
		Suppose
		\begin{equation}	\label{1}
		\nu (x,\infty) ~\sim~ x^{-\alpha}\ell(x),	\quad	x \to \infty,
		\end{equation}
		for some $\alpha\in (1,2)$ and some $\ell$ slowly varying at $\infty$. Then
		\begin{eqnarray*}
		\lim_{s \to \infty} \frac{\E |T(s)-{\tt m}^{-1}s|}{c(s)}
		& = &
		{\tt m}^{-(\alpha+1)/\alpha} \E |W|	\\
		& = &
		\frac{2}{\pi{\tt m}^{(\alpha+1)/\alpha}}\Gamma(1-1/\alpha)\Gamma(1-\alpha)^{1/\alpha} \sin{(\pi/\alpha)}
		\end{eqnarray*}
		where $c(x)$ is a positive function such that $\lim_{x \to \infty} x\ell(c(x))c^{-\alpha}(x)=1$,
		and $W$ is a random variable with characteristic function \eqref{eq:characteristic function}.
\end{itemize}
\end{Theorem}


\section{Proofs}	\label{sec:proofs}

\begin{proof}[Proof of Theorem \ref{Thm:moment convergence}]
It is known (see, for instance, Theorem 3b in \cite{Bingham:1972})
that the following convergence in distribution takes place:
\begin{equation}	\label{eq:distributional convergence of N}
\frac{N(s)-\mu^{-1}s}{g(s)}	~\stackrel{\mathrm{d}}{\to}~	W, \quad s \to \infty,
\end{equation}
where
\begin{itemize}
	\item
		in the case (A1) $W$ is a random variable with standard normal law and $g(s):=\sqrt{\sigma^2\mu^{-3}s}$;
	\item
		in the case (A2) $W$ is a random variable with standard normal law and $g(s):=\mu^{-3/2}c(s)$
		with $c(x)$ being a positive function satisfying $\lim_{x \to \infty} x \ell(c(x))/c(x)^2 = 1$;
	\item
		in the case (A3) $W$ is a random variable with characteristic function given by \eqref{eq:characteristic function}
		and $g(s):=\mu^{-(1+\alpha)/\alpha}c(s)$ where $c(x)$ is a positive function such that
		$\lim_{x \to \infty} x \ell(c(x))/c(x)^{\alpha} = 1$.
\end{itemize}
Putting $c(x) := \sigma \sqrt{x}$ in the case (A1) we observe that
$c(x)$ is regularly varying at $\infty$ with index $1/\alpha$,
where $\alpha=2$ in the cases (A1) and (A2).
We only check this for the case (A2), the case (A3) being similar, and the case (A1) being trivial.

The function $c(x)$ is an asymptotic inverse of
$x^2/\int_{[0,\,x]} y^2 \, \Prob\{\xi\in \dy\} \sim x^2/\ell(x)$.
Hence, by Proposition 1.5.15 in \cite{BGT}, $c(x) \sim x^{1/2}(L^\#(x))^{1/2}$
where $L^\#(x)$ denotes the de Bruijn conjugate of $L(x)=1/\ell(x^{1/2})$.

Our purpose is to show that
\begin{equation}	\label{eq:moment convergence}
\lim_{s \to \infty} \frac{\E |N(s)-\mu^{-1}s|}{g(s)}	~=~	\E |W|.
\end{equation}
%We first prove \eqref{eq:moment convergence} when $s$ goes to $\infty$ along the
%sequence $\big(\mu n\big)_{n\in\N}$.
We start with the representation
\begin{eqnarray*}
\E |S_{N(\mu n)}-S_n|
& = &
\E(S_{N(\mu n)\vee n}-S_{N(\mu n) \wedge n})	\\
& = &
\mu\E\big((N(\mu n)\vee n) - (N(\mu n)\wedge n) \big) ~=~ \mu \E \big|N(\mu n)-n|,
\end{eqnarray*}
where the second equality follows from Wald's identity.
We thus infer
\begin{eqnarray}
\E \big( |S_n-\mu n| - (S_{N(\mu n)} - \mu n) \big)
& \leq &
\mu \E |N(\mu n)-n|	\notag	\\
& \leq &
\E \big(|S_n-\mu n| +(S_{N(\mu n)}-\mu n)\big).	\label{eq:bounds on E|N(mun)-n|}
\end{eqnarray}
It is a classical result that
\begin{equation}	\label{xxv}
{S_n-\mu n\over c(n)} \ \stackrel{\mathrm{d}}{\to} \ -W, \quad n\to\infty.
\end{equation}
Furthermore, according to Lemma 5.2.2 in \cite{Ibr},
\begin{equation*}
\underset{n\in\N}{\sup}\,\E \bigg({\big|S_n-\mu n\big|\over c(n)}\bigg)^{1+\delta}	~<~	\infty
\end{equation*}
for some $\delta>0$.
This together with \eqref{xxv} ensures that
\begin{equation}\label{xxu}
\lim_{n \to \infty} {\E \big|S_n-\mu n\big|\over c(n)}	~=~	\E |W|.
\end{equation}
From \cite{Mohan} it is known that as $s\to\infty$,
\begin{equation*}
\E(S_{N(s)}-s)	~\sim~	\begin{cases}
						\text{const}						&	\text{in the case (A1),}		\\
						\text{const}\,\ell(s)					&	\text{in the case (A2),}		\\
						\text{const}\,s^{2-\alpha} \ell(s)	&	\text{in the case (A3),}
						\end{cases}
\end{equation*}
provided that the law of $\xi$ is non-lattice.

Assume now that the law of $\xi$ is lattice with span $d>0$.
In the case (A1), according to Theorem 9 in \cite{Feller},
\begin{equation*}
\lim_{n \to \infty} \E(S_{N(nd)}-nd)	~\to~	\text{const}.
\end{equation*}
Hence
\begin{equation*}
\E (S_{N(s)}-s)	~=~	O(1) \quad \text{as } s \to \infty.
\end{equation*}
In the cases (A2) and (A3), according to Theorem 6 in \cite{Sg},
$\E(S_{N(s)}-s)$ exhibits the same asymptotics as in the non-lattice case.

Recalling that $c(s)$ is regularly varying at $\infty$ with index
$1/\alpha$ (where $\alpha=2$ in the Cases (A1) and (A2)), we conclude that
\begin{equation*}
\lim_{t \to \infty} \frac{\E S_{N(\mu n)}-\mu n}{c(n)} = 0.
\end{equation*}
Applying this and \eqref{xxu} to \eqref{eq:bounds on E|N(mun)-n|}
we infer
\begin{equation*}
\lim_{n \to \infty} \mu \frac{\E |N(\mu n)-n|}{c(n)}	~=~	\E |W|.
\end{equation*}

Now we have to check that this relation implies \eqref{eq:moment convergence}.
For any $s>0$ there exists $n=n(s)\in\N_0$ such that $s\in (\mu n,\mu(n+1)]$.
Hence, by subadditivity,
\begin{equation*}
\E \big(N(s)-N(\mu n)\big)	~\leq~	\E \big(N(\mu(n+1))-N(\mu n)\big)	~\leq~	\E N(\mu).
\end{equation*}
It remains to observe that the regular variation of $c(s)$ entails
$\lim_{s \to \infty} {c(\mu n(s)\mu^{-1})\over c(s)}=\mu^{-1/\alpha}$,
hence \eqref{eq:moment convergence}.

The formula for $\E |W|$ in the case (A3) is proved in Lemma \ref{Lem:mean}. The proof is complete.
\end{proof}


\begin{proof}[Proof of Theorem \ref{Thm:moment convergence subordinator}]
In the proof of Lemma 6(a) in \cite{Gol}, it is shown that
conditions \eqref{2} entail
\begin{equation*}
\int_{[0,\,x]}y^2\Prob\{S(1)\in \dy\} ~\sim~ \ell(x),	\quad x \to \infty.
\end{equation*}
According to Proposition 0 in the same reference, condition \eqref{1} entails
\begin{equation*}
\Prob\{S(1)>x\} \ \sim \ x^{-\alpha}\ell(x), \quad x \to \infty.
\end{equation*}
Let $N^*(s) := \#\{k \in \N_0: S(k) \leq s\}$, $s \geq 0$.
According to Theorem \ref{Thm:moment convergence},
\begin{equation*}
\lim_{s \to \infty} \frac{\E |N^*(s)-{\tt m}^{-1}s|}{g(s)}	~=~	\E |W|
\end{equation*}
where in the case (B1)
$g(s)=\sqrt{{\tt s}^2{\tt m}^{-3}s}$, in the case (B2) $g(s) = {\tt m}^{-3/2}c(s)$,
and in the case (B3) $g(s)={\tt m}^{-(1+\alpha)/\alpha}c(s)$.
It remains to note that, for $s\geq 0$, $N^*(s)-T(s) \in [0,1]$ a.s.
The proof is complete.
\end{proof}


\section{Auxiliary result}

\begin{Lemma}	\label{Lem:mean}
Let $W$ be a random variable with characteristic function given by \eqref{eq:characteristic function}.
Then, for $r < \alpha$,
\begin{equation*}
\E |W|^r ~=~
\frac{2\Gamma(r+1)}{\pi r} \sin{(r\pi/2)} \Gamma(1-r/\alpha)\Gamma(1-\alpha)^{r/\alpha} \cos{(\pi r/2-\pi r/\alpha)}.
\end{equation*}
In particular,
\begin{equation*}
\E |W|	~=~	2\pi^{-1}\Gamma(1-1/\alpha)\Gamma(1-\alpha)^{1/\alpha} \sin{(\pi/\alpha)}.
\end{equation*}
\end{Lemma}
\begin{proof}
We use the integral representation for the $r$th absolute moment
(see Lemma 2 in \cite{BE})
\begin{equation}	\label{eq:integral representation}
m_r	~:=~	\E|W|^r
~=~ \frac{\Gamma(r+1)}{\pi} \sin \Big(\frac{r \pi}{2}\Big)
\int_{\R} \frac{1-{\rm Re}\,\E e^{{\rm i} t W}}{|t|^{r+1}} \, \dt.
\end{equation}
Set $A:=\pi^{-1}\Gamma(r+1)\sin{(r\pi/2)}$, $B:=\Gamma(1-\alpha)\cos{(\pi\alpha/2)}$ and
$C:=\Gamma(1-\alpha)\sin{(\pi\alpha/2)}$.
Using Euler's identity $\exp{{\rm i}x}=\cos x +{\rm i}\sin x$ in \eqref{eq:characteristic function}, we obtain
\begin{equation*}
{\rm Re}\,\E e^{{\rm i}tW}
~=~	\exp{(-B|t|^{\alpha})}\cos{(-C|t|^{\alpha}{\rm sgn}(t))}.
\end{equation*}
Substituting this into formula \eqref{eq:integral representation} yields
\begin{equation*}
m_r
~=~
2A\int_0^{\infty} \frac{1-\exp{(-Bt^{\alpha})}\cos{(Ct^{\alpha})}}{t^{r+1}} \, \dt.
\end{equation*}
A change of variables ($u := t^{\alpha}$) gives
\begin{eqnarray}
m_r
&=&
\frac{2A}{\alpha}\int_0^{\infty}\big(1-\exp{(-Bu)}\cos{(Cu)}\big) u^{-1-r/\alpha} \, \du	\notag	\\
&=&
\frac{2A}{\alpha} \int_0^{\infty}\big(1-\exp{(-Bu)}\big) u^{-1-r/\alpha} \, \du	\notag	\\
& &
+ \frac{2A}{\alpha} \int_0^{\infty}\big(\exp{(-Bu)}-\exp{(-Bu)}\cos{(Cu)}\big) u^{-1-r/\alpha} \, \du	\notag	\\
&=:&
I_1+I_2.	\label{eq:I_1+I_2}
\end{eqnarray}
According to formula (3.945(2)) in \cite{GradRyj}, we have
\begin{equation*}
I_2	~=~	\frac{2A}{\alpha} \, \Gamma(-r/\alpha)\Big(B^{r/\alpha}-\Gamma(1-\alpha)^{r/\alpha} \cos{(\pi r/2-\pi r/\alpha)}\Big).
\end{equation*}
To calculate $I_1$ we use integration by parts:
\begin{eqnarray*}
I_1
& = &
\frac{2A}{\alpha}\int_0^{\infty}\big(1-\exp{(-Bu)}\big) u^{-1-r/\alpha} \, \du	\\
& = &
\frac{2AB}{r}\int_0^{\infty}u^{-r/\alpha}\exp{(-Bu)} \du	\\
& = &
\frac{2AB^{r/\alpha}}{r} \Gamma(1-r/\alpha)
~=~ -\frac{2AB^{r/\alpha}}{\alpha} \Gamma(-r/\alpha).
\end{eqnarray*}
Now plugging in the values of $I_1$ and $I_2$ in \eqref{eq:I_1+I_2} gives
\begin{eqnarray*}
m_r
& = &
-\frac{2A}{\alpha}\Gamma(-r/\alpha)\Gamma(1-\alpha)^{r/\alpha} \cos{(\pi r/2-\pi r/\alpha)}	\\
& = &
\frac{2A}{r}\Gamma(1-r/\alpha)\Gamma(1-\alpha)^{r/\alpha} \cos{(\pi r/2-\pi r/\alpha)}.
\end{eqnarray*}
The proof is complete.
\end{proof}

\begin{thebibliography}{99}
\footnotesize

\bibitem{BE} {\sc von Bahr, B. and Esseen, C.-G.} (1965). Inequalities for the $r$th absolute moment of a sum of random variables, $1\leq r\leq 2$.
{\em Ann. Math. Statist.} {\bf 36}, 299--303.

\bibitem{Bingham:1972}{\sc Bingham, N.~H.} (1972). Limit theorems for regenerative phenomena, recurrent events and renewal theory.
{\em Z. Wahrscheinlichkeitstheorie verw. Geb.} {\bf 21}, 20--44.

\bibitem{BGT}{\sc Bingham, N.~H., Goldie, C.~M., and Teugels, J.~L.} (1989).
{\it Regular variation}. Cambridge University Press: Cambridge.

\bibitem{Gol} {\sc Embrechts, P. and Goldie, C.~M.} (1981). Comparing the tail of an infinitely divisible distribution with integrals
of its L\'{e}vy measure. {\em Ann. Probab.} {\bf 9}, 468--481.

\bibitem{Feller} {\sc Feller, W.} (1949). Fluctuation theory of
recurrent events. {\em Trans. Amer. Math. Soc.} {\bf 67}, 98--119.

%\bibitem{Feller:1971} {\sc Feller, W.} (1971). {\it An Introduction to Probability Theory and its Applications.} Volume II. John Wiley \& Sons, New York.

\bibitem{GneIks}{\sc Gnedin, A. and Iksanov, A.} (2012+). Regenerative compositions in the case of slow variation:
A renewal theory approach. Preprint available at
http://arxiv.org/abs/1109.5985.

\bibitem{GradRyj} {\sc Gradstein, I.~ S. and Ryzhik, I.~M.} (2000). {\it Tables of integrals, series and products}. Translated from the Russian.
Sixth edition, Academic Press, Inc., San Diego.

\bibitem{Gut} {\sc Gut, A.} (2009). {\it Stopped random walks. Limit theorems and
applications}. 2nd edition, Springer-Verlag, New York.

\bibitem{Ibr} {\sc Ibragimov, I.~A. and Linnik, Yu.~ V.} (1971). {\it Independent and stationary
sequences of random variables}. Wolters-Noordhoff Publishing:
Groningen.

\bibitem{IMM} {\sc Iksanov, A., Marynych, A. and  Meiners, M.} (2012+). Limit theorems for renewal shot noise processes.
In preparation.

\bibitem{Mohan} {\sc Mohan, N.~R.} (1976). Teugels' renewal
theorem and stable laws. {\em Ann. Probab.} {\bf 4}, 863--868.

\bibitem{Sg} {\sc Sgibnev, M.~S.} (1981). Renewal theorem in the
case of an infinite variance. {\em Siberian Math. J.} {\bf 22},
787--796.

\end{thebibliography}
\end{document}

1208.3964.extracted_bib
\begin{thebibliography}{99}
\footnotesize

\bibitem{BE} {\sc von Bahr, B. and Esseen, C.-G.} (1965). Inequalities for the $r$th absolute moment of a sum of random variables, $1\leq r\leq 2$.
{\em Ann. Math. Statist.} {\bf 36}, 299--303.

\bibitem{Bingham:1972}{\sc Bingham, N.~H.} (1972). Limit theorems for regenerative phenomena, recurrent events and renewal theory.
{\em Z. Wahrscheinlichkeitstheorie verw. Geb.} {\bf 21}, 20--44.

\bibitem{BGT}{\sc Bingham, N.~H., Goldie, C.~M., and Teugels, J.~L.} (1989).
{\it Regular variation}. Cambridge University Press: Cambridge.

\bibitem{Gol} {\sc Embrechts, P. and Goldie, C.~M.} (1981). Comparing the tail of an infinitely divisible distribution with integrals
of its L\'{e}vy measure. {\em Ann. Probab.} {\bf 9}, 468--481.

\bibitem{Feller} {\sc Feller, W.} (1949). Fluctuation theory of
recurrent events. {\em Trans. Amer. Math. Soc.} {\bf 67}, 98--119.

%\bibitem{Feller:1971} {\sc Feller, W.} (1971). {\it An Introduction to Probability Theory and its Applications.} Volume II. John Wiley \& Sons, New York.

\bibitem{GneIks}{\sc Gnedin, A. and Iksanov, A.} (2012+). Regenerative compositions in the case of slow variation:
A renewal theory approach. Preprint available at
http://arxiv.org/abs/1109.5985.

\bibitem{GradRyj} {\sc Gradstein, I.~ S. and Ryzhik, I.~M.} (2000). {\it Tables of integrals, series and products}. Translated from the Russian.
Sixth edition, Academic Press, Inc., San Diego.

\bibitem{Gut} {\sc Gut, A.} (2009). {\it Stopped random walks. Limit theorems and
applications}. 2nd edition, Springer-Verlag, New York.

\bibitem{Ibr} {\sc Ibragimov, I.~A. and Linnik, Yu.~ V.} (1971). {\it Independent and stationary
sequences of random variables}. Wolters-Noordhoff Publishing:
Groningen.

\bibitem{IMM} {\sc Iksanov, A., Marynych, A. and  Meiners, M.} (2012+). Limit theorems for renewal shot noise processes.
In preparation.

\bibitem{Mohan} {\sc Mohan, N.~R.} (1976). Teugels' renewal
theorem and stable laws. {\em Ann. Probab.} {\bf 4}, 863--868.

\bibitem{Sg} {\sc Sgibnev, M.~S.} (1981). Renewal theorem in the
case of an infinite variance. {\em Siberian Math. J.} {\bf 22},
787--796.

\end{thebibliography}