\documentclass[12pt]{article} \usepackage{e-jc} \specs{P3}{20(3)}{2013} \usepackage{amsthm,amsmath,amssymb} \usepackage{graphicx} \newcommand{\E}{{\bf{E}}} \newcommand{\PP}{{\bf{P}}} \newcommand{\Var}{{\bf{Var}}} \newcommand{\Bin}{{\bf{Bin}}} % use this command to create hyperlinks (optional and recommended) \usepackage[colorlinks=true,citecolor=black,linkcolor=black,urlcolor=blue]{hyperref} % use these commands for typesetting doi and arXiv references in the bibliography \newcommand{\doi}[1]{\href{http://dx.doi.org/#1}{\texttt{doi:#1}}} \newcommand{\arxiv}[1]{\href{http://arxiv.org/abs/#1}{\texttt{arXiv:#1}}} % all overfull boxes must be fixed; % i.e. there must be no text protruding into the margins % declare theorem-like environments \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{fact}[theorem]{Fact} \newtheorem{observation}[theorem]{Observation} \newtheorem{claim}[theorem]{Claim} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{open}[theorem]{Open Problem} \newtheorem{problem}[theorem]{Problem} \newtheorem{question}[theorem]{Question} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{note}[theorem]{Note} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % if needed include a line break (\\) at an appropriate place in the title \title{\bf Degree distribution of an inhomogeneous\\ random intersection graph} % input author, affilliation, address and support information as follows; % the address should include the country, and does not have to include % the street address \author{Mindaugas Bloznelis\thanks{Supported by the Research Council of Lithuania Grant MIP-053/2011.}\\ \small Faculty of mathematics and informatics\\[-0.8ex] \small Vilnius University\\[-0.8ex] \small Vilnius, Lithuania\\ \small\tt mindaugas.bloznelis@mif.vu.lt\\ \and Julius Damarackas\\ \small Faculty of mathematics and informatics\\[-0.8ex] \small Vilnius University\\[-0.8ex] \small Vilnius, Lithuania\\ \small\tt julius.damarackas@mif.stud.vu.lt } % \date{\dateline{submission date}{acceptance date}\\ % \small Mathematics Subject Classifications: comma separated list of % MSC codes available from http://www.ams.org/mathscinet/freeTools.html} \date{\dateline{Oct 8, 2012}{Jul 11, 2013}{Jul 19, 2013}\\ \small Mathematics Subject Classifications: 05C80, 05C07, 05C82} \begin{document} \maketitle % E-JC papers must include an abstract. The abstract should consist of a % succinct statement of background followed by a listing of the % principal new results that are to be found in the paper. The abstract % should be informative, clear, and as complete as possible. Phrases % like "we investigate..." or "we study..." should be kept to a minimum % in favor of "we prove that..." or "we show that...". Do not % include equation numbers, unexpanded citations (such as "[23]"), or % any other references to things in the paper that are not defined in % the abstract. The abstract will be distributed without the rest of the % paper so it must be entirely self-contained. \begin{abstract} We show the asymptotic degree distribution of the typical vertex of a sparse inhomogeneous random intersection graph. % keywords are optional \bigskip\noindent \textbf{Keywords:} degree distribution; random graph; random intersection graph; power law \end{abstract} \section{Introduction} Let $X_1,\dots, X_m, Y_1,\dots, Y_n$ be independent non-negative random variables such that each $X_i$ has the probability distribution $P_1$ and each $Y_j$ has the probability distribution $P_2$. Given realized values $X=\{X_i\}_{i=1}^m$ and $Y=\{Y_j\}_{j=1}^n$ we define the random bipartite graph $H_{X,Y}$ with the bipartition $V=\{v_1,\dots, v_n\}$, $W=\{w_1,\dots, w_m\}$, where edges $\{w_i,v_j\}$ are inserted with probabilities $p_{ij}=\min\{1, X_iY_j(nm)^{-1/2}\}$ independently for each $\{i,j\}\in [m]\times [n]$. The {\it inhomogeneous} random intersection graph $G(P_1,P_2, n,m)$ defines the adjacency relation on the vertex set $V$: vertices $v',v''\in V$ are declared adjacent (denoted $v'\sim v''$) whenever $v'$ and $v''$ have a common neighbour in $H_{X,Y}$. The degree distribution of the typical vertex of the random graph $G(P_1,P_2, n,m)$ has been first considered by Shang \cite{Shang2010}. The proof of the main result of \cite{Shang2010} contains gaps and the result is incorrect in the regime where $m/n\to\beta\in (0,+\infty)$ as $m,n\to+\infty$. We remark that this regime is of particular importance, because it leads to inhomogeneous graphs with the clustering property: the clustering coefficient $\PP(v_1\sim v_2|v_1\sim v_3, v_2\sim v_3)$ is bounded away from zero provided that $0<\beta<+\infty$ and $\E X_1^3<\infty$, $\E Y_1^2<\infty$, see \cite{BloznelisKurauskas2012}. The aim of the present paper is to show the asymptotic degree distribution in the case where $m/n\to\beta$ for some $\beta\in [0,+\infty]$. We consider a sequence of graphs $\{G_n=G(P_1,P_2, n,m)\}$, where $m=m_n\to +\infty$ as $n\to \infty$, and where $P_1, P_2$ do not depend on $n$. We denote $a_k=\E X_1^k$, $b_k=\E Y_1^k$. By $d(v)=d_{G_n}(v)$ we denote the degree of a vertex $v$ in $G_n$ (the number of vertices adjacent to $v$ in $G_n$). We remark that for every $n$ the random variables $d_{G_n}(v_1),\dots, d_{G_n}(v_n)$ are identically distributed. In Theorem \ref{T1} below we show the asymptotic distribution of $d(v_1)$. \begin{theorem} \label{T1} Let $m,n\to\infty$. (i) Assume that $m/n\to 0$. Suppose that that $\E X_1<\infty$. Then $\PP(d(v_1)=0)=1-o(1)$. (ii) Assume that $m/n\to \beta$ for some $\beta\in (0,+\infty)$. Suppose that $\E X_1^2<\infty$ and $\E Y_1<\infty$. Then $d(v_1)$ converges in distribution to the random variable \begin{equation}\label{d*1} d_*=\sum_{j=1}^{\Lambda_1}\tau_j, \end{equation} where $\tau_1,\tau_2,\dots$ are independent and identically distributed random variables independent of the random variable $\Lambda_1$. They are distributed as follows. For $r=0,1,2,\dots$, we have \begin{equation}\label{d*1++} \PP(\tau_1=r)=\frac{r+1}{\E\Lambda_2}\PP(\Lambda_2=r+1) \qquad {\text{and}} \qquad \PP(\Lambda_i=r)=\E \,e^{-\lambda_i}\frac{\lambda_i^r}{r!}, \qquad i=1,2. \end{equation} Here $\lambda_1=Y_1a_1\beta^{1/2}$ and $\lambda_2=X_1b_1\beta^{-1/2}$. (iii) Assume that $m/n\to+\infty$. Suppose that $\E X_1^2<\infty$ and $\E Y_1<\infty$. Then $d(v_1)$ converges in distribution to a random variable $\Lambda_3$ having the probability distribution \begin{equation}\label{d*2} \PP(\Lambda_3=r)=\E e^{-\lambda_3}\frac{\lambda_3^r}{r!}, \qquad r=0,1,\dots. \end{equation} Here $\lambda_3=Y_1a_2b_1$. \end{theorem} \begin{remark}\label{Remark1} The probability distributions $P_{\Lambda_i}$ of $\Lambda_i$, $i=1,2,3$, are Poisson mixtures. One way to sample from the distribution $P_{\Lambda_i}$ is to generate random variable $\lambda_i$ and then, given $\lambda_i$, to generate Poisson random variable with the parameter $\lambda_i$. The realized value of the Poisson random variable has the distribution $P_{\Lambda_i}$. \end{remark} \begin{remark}\label{Remark2} The asymptotic distributions (\ref{d*1}) and (\ref{d*2}) admit heavy tails. In the case (ii) we obtain a power law asymptotic degree distribution (\ref{d*1}) provided that the heavier of the tails $t\to\PP(\tau_1>t)$ and $t\to\PP(\Lambda_1>t)$ has a power law, see, e.g., \cite{Foss}. % least one %of $P_1$ and $P_2$ has a power law and $P_1(0),P_2(0)<1$. Similarly, in the case (iii) we obtain a power law asymptotic degree distribution (\ref{d*2}) provided that $P_2$ has a power law. \end{remark} \begin{remark}\label{Remark3} Since the second moment $a_2$ does not show up in (\ref{d*1}), (\ref{d*1++}) we expect that in the case (ii) the second moment condition $\E X_1^2<\infty$ is redundant and could perhaps be replaced by the weaker first moment condition $\E X_1<\infty$. \end{remark} Random intersection graphs have attracted considerable attention in the recent literature, see, e.g., \cite{Barbour2011, behrisch2007, Blackburn2009, Bradonjic2010, Britton2008, eschenauer2002, Rybarczyk2011}. Starting with the paper by Karo\'nski et al \cite{karonski1999}, see also \cite{Singer1995}, where the case of degenerate distributions $P_1=P_{1n}$, $P_2=P_{2n}$ depending on $n$ was considered (i.e., $\PP_{1n}(c_n)=\PP_{2n}(c_n)=1$, for some $c_n>0$), several more complex random intersection graph models were later introduced by Godehardt and Jaworski \cite{godehardt2001}, Spirakis et al. \cite{Spirakis2004}, Shang \cite{Shang2010}. The asymptotic degree distribution for various random intersection graph models was shown in \cite{Bloznelis2008, Bloznelis2010, Bloznelis2010+, Bloznelis2011+, Deijfen, JKS, JaworskiStark2008, Rybarczyk-degree2011, stark2004}. \section{Proof} Before the proof of Theorem \ref{T1} we introduce some notation and give two auxiliary lemmas. The event that the edge $\{w_i, v_j\}$ is present in $H=H_{X,Y}$ is denoted $w_i\to v_j$. We denote \begin{displaymath} {\mathbb I}_{ij}={\mathbb I}_{\{w_i\to v_j\}}, \qquad {\mathbb I}_i={\mathbb I}_{i1}, \qquad u_i=\sum_{2\le j\le n}{\mathbb I}_{ij}, \qquad L=\sum_{i=1}^mu_i{\mathbb I}_i \end{displaymath} and remark that $u_i$ counts all neighbours of $w_i$ in $H$ belonging to the set $V\setminus \{v_1\}$. Denote \begin{eqnarray}\nonumber && {\hat a}_k=m^{-1}\sum_{1\le i\le m}X^k_i, \qquad {\hat b}_k=n^{-1}\sum_{2\le j\le n}Y^k_j, \\ \label{S-XY} && \lambda_{ij}=\frac{X_iY_j}{\sqrt{mn}}, \qquad S_{XY}=\sum_{i=1}^m\sum_{j=2}^n\frac{X_i}{\sqrt{mn}}\lambda_{ij}\min\{1,\lambda_{ij}\}, \end{eqnarray} and introduce the event ${\cal A}_1=\{\lambda_{i1}<1, \, 1\le i\le m\}$. We denote $\beta_n=m/n$. By ${\PP}_1$ and $\E_1$ we denote the conditional probability and conditional expectation given $Y_1$. By ${\tilde \PP}$ and ${\tilde \E}$ we denote the conditional probability and conditional expectation given $X,Y$. By $d_{TV}(\zeta,\xi)$ we denote the total variantion distance between the probability distributions of random variables $\zeta$ and $\xi$. In the case where $\zeta,\xi$, $X, Y$ are defined on the same probability space we denote by ${\tilde d}_{TV}(\zeta,\xi)$ the total variation distance between the conditional distributions of $\zeta$ and $\xi$ given $X,Y$. In the proof below we use the following simple fact about the convergence of a sequence of random variables $\{\varkappa_n\}$: \begin{equation}\label{fact} \varkappa_n=o_P(1), \quad \E \sup_n\varkappa_n<\infty \quad \Rightarrow \quad \E\varkappa_n=o(1). \end{equation} We remark that the condition $\E \sup_n\varkappa_n<\infty$ (which assumes implicitly that all $\varkappa_n$ are defined on the same probability space) can be replaced by a more restrictive condition that there exists a constant $c>0$ such that $|\varkappa_n|\le c$, for all $n$. In the latter case we do not need all $\varkappa_n$ to be defined on the same probability space. In particular, given a sequence of bivariate random vectors $\{(\eta_n,\theta_n)\}$ such that, for every $n$ and $m=m_n$ random variables $\eta_n$, $\theta_n$ and $\{X_i\}_{i=1}^m$, $\{Y_j\}_{j=1}^n$ are defined on the same probability space, we have \begin{displaymath} {\tilde d}_{TV}(\eta_n,\theta_n)=o_P(1) \ \Rightarrow \ d_{TV}(\eta_n,\theta_n)\le \E{\tilde d}_{TV}(\eta_n,\theta_n) =o(1). \end{displaymath} Here and below all limits are taken as $n\to+\infty$ (if not stated otherwise). \begin{lemma} \label{d-L} Assume that $\E X_1^2<\infty$ and $\E Y_1<\infty$. We have as $n\to+\infty$ \begin{eqnarray}\label{d-L21} && \PP(d(v_1)\not=L)=o(1), \\ \label{d-L22} && \PP({\cal A}_1)=1-o(1), \\ \label{d-L23} && S_{XY}=o_P(1), \qquad \E S_{XY}=o(1). \end{eqnarray} \end{lemma} \begin{proof} Proof of (\ref{d-L21}). We observe that the event ${\cal A}=\{d(v_1)\not=L\}$ occurs in the case where for some $2\le j\le n$ and some distinct $i_1,i_2\in [m]$ the event ${\cal A}_{i_1,i_2,j}= \{w_{i_1}\to v_1,\, w_{i_1}\to v_j,\, w_{i_2}\to v_1,\, w_{i_2}\to v_j\}$ occurs. From the union bound and the inequality ${\tilde \PP}({\cal A}_{i_1,i_2,j})\le m^{-2}n^{-2}X_{i_1}^2X_{i_2}^2Y_1^2Y_j^2$ we obtain \begin{equation}\label{d-L1} {\tilde \PP}({\cal A}) = {\tilde \PP} \left(\bigcup_{\{i_1, i_2\}\subset[m]}\bigcup_{2\le j\le n}{\cal A}_{i_1,i_2,j}\right) \le n^{-1}{\hat b}_2Y_1^2 Q_X. \end{equation} Here $Q_X=m^{-2}\sum_{\{i_1,i_2\}\subset[m]}X_{i_1}^2X_{i_2}^2$. We note that $Q_X$ and $Y_1^2$ are stochastically bounded and %We note that the product $Y_1^2Q_X$ is stochastically bounded and $n^{-1}{\hat b}_2=o_P(1)$ %converges to zero in probability as $n\to+\infty$. Therefore, %the random variable ${\tilde \PP}({\cal A})=o_P(1)$. %converges to zero in probability. Now (\ref{fact}) implies (\ref{d-L21}). %This implies (\ref{d-L21}) since, for any $\varepsilon\in (0,1)$, %\begin{displaymath} %\PP({\cal A})=\E ({\tilde \PP}({\cal A})) %\le %\varepsilon+\PP({\tilde \PP}({\cal A})>\varepsilon) %=\varepsilon+o(1). %\end{displaymath} \medskip Proof of (\ref{d-L22}). Let ${\overline {\cal A}}_1$ denote the complement event to ${\cal A}_1$. %Next we show that $\PP({\overline{\cal A}}_1)=o(1)$. %We obtain from We have, by the union bound and Markov's inequality, %that \begin{displaymath} \PP_1({\overline {\cal A}}_1)\le \sum_{i\in [m]}\PP_1(\lambda_{i1}\ge 1) \le (nm)^{-1}Y_1^2\sum_{i\in [m]}\E X_i^2=n^{-1}a_2Y_1^2. \end{displaymath} We conclude that $\PP_1({\overline {\cal A}}_1)=o_P(1)$, and now (\ref{fact}) implies $\PP({\overline {\cal A}}_1)=\E \PP_1({\overline {\cal A}}_1)=o(1)$. %(\ref{d-L22}). %Note that this implies (\ref{d-L22}) since, %for any $\varepsilon\in (0,1)$, we have %\begin{displaymath} % \PP({\overline{\cal A}}_1) %= %\E\PP_1({\overline {\cal A}}_1) %\le %\varepsilon+\PP(\PP_1({\overline {\cal A}}_1)>\varepsilon) %= %\varepsilon+o(1). %\end{displaymath} \medskip Proof of (\ref{d-L23}). Since the first bound of (\ref{d-L23}) follows from the second one, we only prove the latter. %Now, we show that $S_{XY}=o_P(1)$. Denote ${\hat X}_1=\max\{X_1, 1\}$ and ${\hat Y}_1=\max\{Y_1,1\}$. We observe that $\E X_1^2<\infty$, $\E Y_1<\infty$ implies \begin{displaymath} \lim_{t\to+\infty}\E {\hat X}_1^2{\mathbb I}_{\{{\hat X}_1>t\}}=0, \qquad \lim_{t\to+\infty}\E {\hat Y}_1{\mathbb I}_{\{{\hat Y}_1>t\}}=0. \end{displaymath} Hence one can find a strictly increasing function $\varphi:[1,+\infty)\to [0,+\infty)$ with $\lim_{t\to+\infty}\varphi(t)$ $=$ $+\infty$ such that \begin{equation}\label{hat1} \E {\hat X}_1^2\varphi({\hat X}_1)<\infty, \qquad \E {\hat Y}_1\varphi({\hat Y}_1)<\infty. \end{equation} We remark that one can find $\varphi$ which satisfies, in addition, the inequalities \begin{equation}\label{phi} \varphi(t)\varepsilon)=o(1)$, for any $\varepsilon\in (0,1)$. For this purpose we write $\PP(L>\varepsilon)=\E\PP_1(L>\varepsilon)$ and prove that $\PP_1(L>\varepsilon)=o_P(1)$, see (\ref{fact}). Here we estimate $\PP_1(L>\varepsilon)$ using the union bound and Markov's inequality, \begin{displaymath} \PP_1(L>\varepsilon) \le \sum_{1\le k\le m}\PP_1({\mathbb I}_k=1) \le \E_1\sum_{1\le k\le m}\lambda_{k1} = \sqrt{m/n}Y_1\E X_1 = o_P(1). \end{displaymath} {\it The case (ii)}. Before the proof we introduce some notation. Given $X,Y$, let \begin{equation}\label{sequences_xi} \{\Delta_{lk}\}_{k=1}^m, \quad l=1,2,3, \qquad \{\xi_{rk}\}_{k=1}^m, \quad r=1,2,3,4, \quad {\text{and}} \quad \{\xi'_{3k}\}_{k=1}^m, \quad \{\xi'_{4k}\}_{k=1}^m, \end{equation} and $\{\eta_k\}_{k=1}^m$ be sequences of conditionally independent (within each sequence) Poisson random variables with mean values \begin{eqnarray}\nonumber && {\tilde \E}\xi_{1k}=\sum_{2\le j\le n} p_{kj}, \qquad {\tilde \E}\xi_{2k}=\sum_{2\le j\le n} \lambda_{kj}, \qquad {\tilde \E}\xi_{3k}=\frac{1}{\sqrt{\beta_n}}b_1X_k, \qquad {\tilde \E}\xi_{4k}=\frac{1}{\sqrt{\beta}}b_1X_k, \\ \nonumber && {\tilde \E}\Delta_{1k}=\sum_{2\le j\le n} (\lambda_{kj}-p_{kj}), \qquad {\tilde \E}\Delta_{2k}=\frac{1}{\sqrt{\beta_n}}\delta_2 X_k, \qquad {\tilde \E}\Delta_{3k}=\frac{1}{\sqrt{\beta_n}}\delta_3 X_k, \\ \nonumber && {\tilde \E}\xi'_{3k}=\beta_n^{-1/2}bX_k, \qquad {\tilde \E}\xi'_{4k}=\beta' b_1 X_k, \qquad {\tilde \E}\eta_{k}=\lambda_{k1}. \end{eqnarray} Here we denote $b:=\min\{{\hat b}_1, b_1\}$, $\delta_{2}:= {\hat b}_1-b$, $\delta_{3}:=b_1-b$ and $\beta':=\min\{\beta^{-1/2},\beta_n^{-1/2}\}$. We recall that ${\tilde \E}$ denotes the conditional expectation given $X,Y$. We assume that, given $X,Y$, the sequences $\{\xi'_{3k}\}_{k=1}^m$, $\{\Delta_{2k}\}_{k=1}^m$, $\{\Delta_{3k}\}_{k=1}^m$ are conditionally independent, the sequences $\{{\mathbb I}_{k}\}_{k=1}^m$, $\{\xi_{3k}\}_{k=1}^m$ are conditionally independent, the sequences $\{\xi_{1k}\}_{k=1}^m$, $\{\Delta_{1k}\}_{k=1}^m$ are conditionally independent, and the sequence $\{\eta_k\}_{k=1}^m$ is conditionally independent of the sequences (\ref{sequences_xi}). We assume, in addition, that for $1\le k\le m$ \begin{displaymath} \xi_{2k}:=\xi_{1k}+\Delta_{1k}, \qquad \xi_{3k}:=\xi_{4k}'+(\beta_n^{-1/2}-\beta') b_1 X_k, \qquad \xi_{4k}:=\xi_{4k}'+(\beta^{-1/2}-\beta') b_1 X_k. \end{displaymath} % almost surely (one may assume that $\xi_{2k}$ is defined as the sum $\xi_{1k}+\Delta_{1k}$). Furthermore, we suppose that, given $X,Y_1$, the sequence $\{\eta_k\}_{k=1}^m$ is conditionally independent of the subgraph of $H_{X,Y}$ spanned by the vertices $\{v_2,\dots, v_n\}\cup W$. \medskip Next we introduce several random variables which are intermediate between $L$ and $d_*$: \begin{eqnarray}\nonumber && L_0=\sum_{1\le k\le m}\eta_ku_k, \qquad L_r=\sum_{1\le k\le m}\eta_k\xi_{rk}, \qquad r=1,2,3,4, \\ \nonumber && L'_2=\sum_{1\le k\le m}\eta_k(\xi'_{3k}+\Delta_{2k}), \qquad L'_3=\sum_{1\le k\le m}\eta_k(\xi'_{3k}+\Delta_{3k}). \end{eqnarray} \medskip Finally, we denote \begin{eqnarray}\nonumber && f_\tau(t)=\E e^{it\tau_1}, \qquad {\bar f}_{\tau}(t)=\sum_{r\ge 0}e^{itr}{\bar p}_r, \qquad {\bar p}_r = {\bar \lambda}^{-1}\sum_{1\le k\le m}\lambda_{k1}{\mathbb I}_{\{\xi_{4k}=r\}}, \qquad {\bar \lambda}=\sum_{k=1}^m\lambda_{k1}, \\ \nonumber && \delta=({\bar f}_\tau(t)-1){\bar \lambda}-(f_{\tau}(t)-1)\lambda_1, \qquad f(t)=\E_1e^{itd_*}, \qquad {\bar f}(t)={\bar\E}e^{it L_4}. \end{eqnarray} Here ${\bar \E}$ denotes the conditional expectation given $X,Y$ and $\xi_{41},\dots, \xi_{4m}$. We recall that $\E_1$ denotes the conditional expectation given $Y_1$. \medskip Let us prove the statement (ii). In view of (\ref{d-L21}) the random variables $d(v_1)$ and $L$ have the same asymptotic distribution (if any). We shall show that $L$ converges in distribution to (\ref{d*1}). We proceed in two steps. Firstly, we approximate the distribution of $L$ by that of $L_4$ using LeCam's inequality, see Lemma \ref{LeCamLemma}, and a coupling argument. Secondly, we prove the convergence of the characteristic functions $\E e^{itL_4}\to\E e^{itd_*}$. Here and below $i=\sqrt{-1}$ denotes the imaginary unit. %XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX {\it Step 1.} Let us show that $L$ and $L_4$ have the same asymptotic probability distribution (if any). To this aim we prove that \begin{eqnarray}\label{DeltaT} && d_{TV}(L, L_0)=o(1), \qquad d_{TV}(L_0,L_1)=o(1), \\ \label{L-L} && \E|L_1-L_2|=o(1), \qquad L'_2-L'_3=o_P(1), \qquad \E|L_3-L_4|=o(1), \end{eqnarray} and observe that $L_2$ (respectively $L_3$) has the same distribution as $L_2'$ (respectively $L_3'$). %.................................... \quad% Let us prove the first bound of (\ref{DeltaT}). In view of (\ref{fact}) it suffices to show that ${\tilde d}_{TV}(L_0,L)=o_P(1)$. In order to prove the latter bound we apply the inequality \begin{equation}\label{TV1} {\tilde d}_{TV}(L_0, L){\mathbb I}_{{\cal A}_1}\le n^{-1}Y_1^2{\hat a}_2 \end{equation} shown below. We remark that (\ref{TV1}) implies \begin{displaymath} {\tilde d}_{TV}(L_0,L) \le {\tilde d}_{TV}(L_0,L) {\mathbb I}_{{\cal A}_1}+{\mathbb I}_{{\overline{\cal A}}_1} \le n^{-1}Y_1^2{\hat a}_2+{\mathbb I}_{{\overline{\cal A}}_1}=o_P(1). \end{displaymath} Here $n^{-1}Y_1^2{\hat a}_2=o_P(1)$, because $Y_1^2{\hat a}_2$ is stochastically bounded. Furthermore, the bound ${\mathbb I}_{{\overline{\cal A}}_1}=o_P(1)$ follows from (\ref{d-L22}). It remains to prove (\ref{TV1}). % from LeCam's inequality, see Lemma \ref{LeCamLemma} below. We denote ${\bar L}_k=\sum_{l=1}^k{\mathbb I}_lu_l$ $+$ $\sum_{l=k+1}^m\eta_lu_l$ and write, by the triangle inequality, %of the total variation distance, \begin{displaymath} {\tilde d}_{TV}(L_0,L)\le \sum_{k=1}^m{\tilde d}_{TV}({\bar L}_{k-1}, {\bar L}_k). \end{displaymath} Then we estimate ${\tilde d}_{TV}({\bar L}_{k-1}, {\bar L}_k) \le {\tilde d}_{TV}(\eta_k,{\mathbb I}_k)\le (nm)^{-1}Y_1^2X_k^2$. Here the first inequality follows from the properties of the total variation distance. The second inequality follows from Lemma \ref{LeCamLemma} and the fact that on the event ${\cal A}_1$ we have $p_{k1}=\lambda_{k1}$. \medskip \quad% Let us prove the second bound of (\ref{DeltaT}). In view of (\ref{fact}) it suffices to show that ${\tilde d}_{TV}(L_0,L_1)=o_P(1)$. We denote $L^*_k=\sum_{l=1}^k\eta_lu_l+\sum_{l=k+1}^m\eta_l\xi_{1l}$ and write, by the triangle inequality, %of the total variation distance, \begin{equation}\label{dTV4} {\tilde d}_{TV}(L_0,L_1)\le \sum_{k=1}^m{\tilde d}_{TV}(L^*_{k-1}, L^*_k). \end{equation} Here \begin{equation}\label{dTV41} {\tilde d}_{TV}(L^*_{k-1}, L^*_k)\le {\tilde d}_{TV}(\eta_ku_k,\eta_k\xi_{1k}) \le {\tilde \PP}(\eta_k\not=0){\tilde d}_{TV}(u_k,\xi_{1k}). \end{equation} Now, invoking the inequalities ${\tilde \PP}(\eta_k\not=0)=1-e^{-\lambda_{k1}}\le \lambda_{k1}$ and ${\tilde d}_{TV}(u_k,\xi_{1k})\le \sum_{j=2}^np^2_{kj}$, we obtain the desired bound from (\ref{d-L23}), (\ref{dTV4}) and (\ref{dTV41}) \begin{displaymath} {\tilde d}_{TV}(L_0,L_1)\le \sum_{k=1}^m\lambda_{k1}\sum_{j=2}^np^2_{kj} \le Y_1S_{XY}=o_P(1). \end{displaymath} \medskip \quad% Let us prove the first bound of (\ref{L-L}). We observe that \begin{displaymath} |L_2-L_1|=L_2-L_1 = \sum_{1\le k\le m}\eta_k\Delta_{1k} \end{displaymath} and \begin{displaymath} {\tilde \E}\sum_{1\le k\le m}\eta_k\Delta_{1k} = \sum_{1\le k\le m}\lambda_{k1}\sum_{2\le j\le n} (\lambda_{kj}-1){\mathbb I}_{\{\lambda_{kj}>1\}} \le Y_1S_{XY}. \end{displaymath} Hence, we obtain $\E|L_2-L_1|\le \E Y_1\E S_{XY}=o(1)$, see (\ref{d-L23}). \medskip \quad% Let us prove the second bound of (\ref{L-L}). It suffices to show that ${\tilde \E}|L'_2-L'_3|=o_P(1)$. We have \begin{equation}\label{L2-L3} {\tilde \E}|L'_2-L'_3| = (\delta_2+\delta_3)Y_1{\hat a}_2=|{\hat b}_1-b_1|Y_1{\hat a}_2=o_P(1). \end{equation} In the last step we used the fact that $Y_1{\hat a}_2=O_P(1)$ and ${\hat b}_1-b_1=o_P(1)$. %,by the law of large numbers. %Finally, we show that (\ref{L2-L3}) implies %the bound $|L'_2-L'_3|=o_P(1)$. Denoting, for short, $H=|L_2'-L_3'|$ and %$h={\tilde \E}{\mathbb I}_{\{H\ge \varepsilon\}}$ we write, for $\varepsilon\in (0,1)$, %\begin{equation}\label{HHHH} % \PP(H\ge\varepsilon) %= %\E h %= %\E h %({\mathbb I}_{\{{\tilde \E}H\ge\varepsilon^2\}} %+ %{\mathbb I}_{\{{\tilde \E}H<\varepsilon^2\}}). %\end{equation} %Using the simple inequality $h\le 1$ and the inequality, %$h\le \varepsilon^{-1}{\tilde \E}H$, which follows from Markov's inequality, we obtain %\begin{eqnarray}\nonumber % \E h %{\mathbb I}_{\{{\tilde \E}H\ge\varepsilon^2\}} %& %\le %& %\E %{\mathbb I}_{\{{\tilde \E}H\ge\varepsilon^2\}}=\PP({\tilde \E}H\ge\varepsilon^2)=o(1), %\\ %\nonumber %\E h {\mathbb I}_{\{{\tilde \E}H<\varepsilon^2\}} %& %\le %& %\E (\varepsilon^{-1}{\tilde \E}H){\mathbb I}_{\{{\tilde \E}H<\varepsilon^2\}} %< %\varepsilon. %\end{eqnarray} %Invoking these inequalities in (\ref{HHHH}) we obtain $\PP(H\ge \varepsilon)<\varepsilon+o(1)$. %Hence $H=o_P(1)$. \quad% The third bound of (\ref{L-L}) is straightforward \begin{displaymath} \E|L_3-L_4| \le \Bigl|\frac{1}{\sqrt{\beta_n}}-\frac{1}{\sqrt{\beta}}\Bigr| b_1\sum_{1\le k\le m}\E\eta_k X_k = \Bigl|1-\frac{\sqrt{\beta_n}}{\sqrt{\beta}}\Bigr| b_1^2a_2=o(1). \end{displaymath} %XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX {\it Step 2.} Here we show that for every real $t$ we have $\E \Delta(t)=o(1)$, where $\Delta(t)$ $=$ $e^{itL_4}-e^{itd_*}$. To this aim we prove that for any realized value $Y_1$ there exists a positive constant $c=c(t,Y_1)$ such that for every $0<\varepsilon<0.5$ we have \begin{equation}\label{epsilon} \limsup_{n,m\to+\infty}|\E(\Delta(t)|Y_1)|1/\varepsilon$ such that $\PP(\tau_1\ge T)< \varepsilon$. By $c_1, c_2, \dots$ we denote positive numbers which do not depend on $n,m$. We observe that, given $Y_1$, the conditional distribution of $d_*$ is the compound Poisson distribution with the characteristic function $f(t)=e^{\lambda_1(f_{\tau}(t)-1)}$. Similarly, given $X,Y$ and $\xi_{41},\dots, \xi_{4m}$, the conditional distribution of $L_4$ is the compound Poisson distribution with the characteristic function ${\bar f}(t)=e^{{\bar \lambda}({\bar f}_{\tau}(t)-1)}$. In the proof of (\ref{epsilon}) we exploit the convergence ${\bar\lambda}\to\lambda_1$ and ${\bar f}_{\tau}(t)\to f_{\tau}(t)$. In what follows we assume that $m,n$ are so large that $\beta\le 2\beta_n\le 4\beta$. \quad% Let us show (\ref{epsilon}). For this purpose we write \begin{displaymath} \E(\Delta(t)|Y_1)=\E_1\Delta(t)=I_{1}+I_{2}, \qquad I_{1}=\E_1\Delta(t){\mathbb I}_{\cal D}, \qquad I_2=\E_1\Delta(t){\mathbb I}_{\overline{\cal D}} \end{displaymath} and estimate $I_1$ and $I_2$. We have $|I_2|\le 2\PP_1({\overline {\cal D}})=2\PP({\overline {\cal D}})=o(1)$, by the law of large numbers. \quad% Now we show that $I_{1}\le c\,\varepsilon+o(1)$. Combining the identity $\E_1\Delta(t)=\E_1f(t)(e^{\delta}-1)$ with the inequalities $|f(t)|\le 1$ and $|e^{s}-1|\le |s|e^{|s|}$ we obtain \begin{equation}\label{2013-05-08} |I_1|\le \E_1|\delta|e^{|\delta|}{\mathbb I}_{\cal D} \le c_1\E_1|\delta|{\mathbb I}_{\cal D}. \end{equation} Here we estimated $e^{|\delta|}\le e^{8\lambda_1}=:c_1$ using the inequalities \begin{displaymath} |\delta|\le 2{\bar \lambda}+2\lambda_1, \qquad {\bar \lambda}=Y_1\beta_n^{1/2}{\hat a}_1\le 3\lambda_1. \end{displaymath} We remark that the last inequality holds for $\beta_n\le 2\beta$ provided that the event ${\cal D}$ occurs. We shall show below that \begin{equation}\label{2013-05-08+1} \E_1|\delta|{\mathbb I}_{\cal D}\le (c_2+\lambda_1c_3+\lambda_1c_4)\varepsilon+o(1). \end{equation} This inequality together with (\ref{2013-05-08}) yields the desired upper bound for $I_1$. \quad% Let us prove (\ref{2013-05-08+1}). We write \begin{displaymath} \delta = ({\bar f}_{\tau}(t)-1)({\bar\lambda}-\lambda_1)+({\bar f}_{\tau}(t)-f_{\tau}(t))\lambda_1 \end{displaymath} and estimate \begin{displaymath} |\delta|\le 2|{\bar\lambda}-\lambda_1|+\lambda_1|{\bar f}_{\tau}(t)-f_{\tau}(t)|. \end{displaymath} Now, from the inequalities $|{\hat a}_1-a_1|<\varepsilon$ and $\beta_n\le 2\beta$ we obtain \begin{displaymath} |{\bar\lambda}-\lambda_1| \le Y_1|{\hat a}_1-a|\beta_n^{1/2} +Y_1a_1|\beta_n^{1/2}-\beta^{1/2}| \le 2Y_1\beta^{1/2}\varepsilon+o(1). \end{displaymath} Hence, $\E_1|{\bar\lambda}-\lambda_1|{\mathbb I}_{\cal D}\le c_2\varepsilon+o(1)$, where $c_2=2Y_1\beta^{1/2}$. We complete the proof of (\ref{2013-05-08+1}) by showing that \begin{equation}\label{2013-05-06} \E_1|{\bar f}_{\tau}(t)-f_{\tau}(t)|{\mathbb I}_{\cal D}\le (c_3+c_4)\varepsilon+o(1). \end{equation} In order to prove (\ref{2013-05-06}) we split \begin{displaymath} {\bar f}_{\tau}(t)-f_{\tau}(t)=\sum_{r\ge 0}e^{itr}({\bar p}_r-p_r)=R_1-R_2+R_3, \end{displaymath} and estimate separately the terms \begin{displaymath} R_1=\sum_{r\ge T}e^{itr}{\bar p}_r, \qquad R_2=\sum_{r\ge T}e^{itr}p_r, \qquad R_3=\sum_{0\le r