Approximation of a random sum of random variables (infinitely divisible distribution) by a triangular array

We know that a Poisson distribution can be approximated by a binomial distribution. More exactly, let $$(X_{jn})_{1\leq j \leq n}$$ be a i.i.d. triangular array such that $$P[X_{jn}= 1 ] = p_n = 1- P[X_{jn}=0]$$ and:

1. $$p_n \to 0$$ as $$n \to \infty$$;
2. $$np_n \to \lambda$$ as $$n \to \infty$$

So we have the following convergence in distribution: $$S_n = \sum_{j=1}^n X_{jn} \overset{d}{\to} N= \sum_{j=1}^N 1, \quad N \sim \hbox{Poisson}(\lambda)$$ Thus, if we want approximate $$N\sim \hbox{Poisson}(\lambda)$$ by a binomial distribution, we can set $$X_{jn} \sim \hbox{Bernoulli}(\lambda/n)$$ or $$S_n \sim \hbox{Binomial}(\lambda/n , n)$$

Now, given $$(\xi_j)_{j=1}^\infty$$ a i.i.d. sequence of random variables independent of $$N$$. Consider $$Y = \sum_{j=1}^N \xi_j$$. Something tells me that I can find a sum $$S_n$$ that converges in distribution to $$Y$$: $$S_n \overset{d}{\to} Y = \sum_{j=1}^N \xi_j, \quad N \sim \hbox{Poisson}(\lambda)$$ Since $$Y$$ is infinitely divisible, so it there is a triangular array $$(X_{ij})$$ such that $$S_n$$ converges in distribution to $$Y$$, But I think I should adjust some weights in the summations: $$S_n= \sum_{j=1}^n w_j X_{jn}$$, where $$X_{jn} \sim \hbox{Bernoulli}(\lambda/n)$$.

Is there any constructive way to express this sum $$S_n$$?

$$\newcommand\la\lambda$$Let $$S_n:=\sum_{j=1}^n\xi_j X_{j,n},$$ where the $$\xi_j$$'s are iid random variables (r.v.'s) and, for each $$n$$, the $$X_{j,n}$$'s are iid r.v.'s independent of $$\xi_j$$'s and such that each $$X_{j,n}$$ has the Bernoulli distribution with parameter $$p_n$$. Suppose that $$n\to\infty$$ and $$np_n\to\la$$ for some real $$\la>0$$.

Then $$\begin{equation*} S_n\to Y:=\sum_{j=1}^N\xi_j \tag{1}\label{1} \end{equation*}$$ in distribution, where $$N$$ is a Poisson r.v. with parameter $$\la$$ independent of the $$\xi_j$$'s.

This follows easily by the method of characteristic functions: If $$f(t):=Ee^{it\xi_1}$$ for real $$t$$, then $$\begin{equation*} Ee^{itS_n}=(Ee^{it\xi_1 X_{1,n}})^n =(1-p_n+p_n f(t))^n\to e^{\la(f(t)-1)} \tag{2}\label{2} \end{equation*}$$ and $$\begin{equation*} Ee^{itY}=\sum_{n=0}^\infty P(N=n)f(t)^n =\sum_{n=0}^\infty \frac{\la^n}{n!}\,e^{-\la}f(t)^n =e^{\la(f(t)-1)}, \tag{3}\label{3} \end{equation*}$$ so that $$Ee^{itS_n}\to Ee^{itY}$$.

The result you quoted in the beginning of your post is the special case of \eqref{1} with $$\xi_j=1$$ for all $$j$$.

Details on \eqref{2}: \begin{equation*} \begin{aligned} Ee^{it\xi_1 X_{1,n}}&=Ee^{it\xi_1 X_{1,n}}\,1(X_{1,n}=0)+Ee^{it\xi_1 X_{1,n}}\,1(X_{1,n}=1) \\ &=E1(X_{1,n}=0)+Ee^{it\xi_1}\,1(X_{1,n}=1) \\ &=1-p_n+Ee^{it\xi_1}\,E1(X_{1,n}=1) \\ &=1-p_n+f(t) p_n. \end{aligned} \end{equation*} Here we used the equality $$Ee^{it\xi_1}\,1(X_{1,n}=1)=Ee^{it\xi_1}\,E1(X_{1,n}=1)$$, which holds because $$\xi_1$$ and $$X_{1,n}$$ are independent.

Details on \eqref{3}: \begin{equation*} \begin{aligned} Ee^{itY}&=\sum_{n=0}^\infty E1(N=n)e^{itY} \\ &=\sum_{n=0}^\infty E1(N=n)\exp\Big(it\sum_{j=1}^n\xi_j\Big) \\ &=\sum_{n=0}^\infty E1(N=n)\,E\exp\Big(it\sum_{j=1}^n\xi_j\Big) \\ &=\sum_{n=0}^\infty P(N=n)f(t)^n. \end{aligned} \end{equation*} Here we used the equality $$E1(N=n)\,\exp\big(it\sum_{j=1}^n\xi_j\big)=E1(N=n)\,E\exp\big(it\sum_{j=1}^n\xi_j\big)$$, which holds because the $$\xi_j$$'s and $$N$$ are independent.

• Let me denote $f_{S_n}(t)= \left(E\left[ e^{it \xi_1 X_{i,n}} \right]\right)^n$. For the following, I will suppose $\xi_j$'s non-random: $$f_{S_n}(t)= ( (1- p_n)e^{it\xi_1 0} + p_ne^{it\xi_1} )^n =(1-p_n + p_n e^{it\xi_1})^n$$ So I have two question: (i) Why $f_{S_n}(t)=(1-p_n + p_n f(t))^n$ and (ii) Are you considering the $\xi_j$'s to be non-random? Would then $S_n$ be constructed from a fixed realization of $(\xi_j)_{j=1}^\infty$?
– Fam
Commented Dec 19, 2022 at 20:33
• @Fam : (i) I have provided details on (2) (and also on (3)). (ii) No, I do not assume that the $\xi_j$'s are nonrandom. All the assumptions are explicitly stated in the answer, and no other assumptions are used. Commented Dec 19, 2022 at 21:04
• Thank you very much. I asked about the non-random, because it seems that this is the case of a more general case, where I compound stochastic processes instead of random variables. Thanks a lot if you can help me: mathoverflow.net/questions/436906/…
– Fam
Commented Dec 20, 2022 at 7:36