Moderate deviations principle for products of sums of random variables (Q547408)

In this paper, asymptotic properties of product \(\prod^n_{k=1} S_k\) of the sums \(S_k\) of positive random variables are investigated. In the introduction, the brief history of the topic is provided. The main result of the paper is the moderate deviation principle presented below. Let \(X_n\), \(n= 1,2,\dots\), be independent identically distributed random variables defined on a probability space \((\Omega,{\mathcal F},P)\), such that \(Ee^{\delta X_1}< \infty\) and \(Ee^{\delta|\ln X_1|}< \infty\) for some number \(\delta> 0\) and \(\mu= EX_1> 0\). Denote \(\sigma^2= E(X_1- EX_1)^2\), \(\gamma= \sigma/\mu\), \(S_n= X_1+\cdots+ X_n\). Let \(\{b_n\}_{n\geq 1}\) be a sequence of positive numbers such that \[ \lim_{n\to\infty}\, {b_n\ln n\over\sqrt{n}}= 0,\quad \lim_{n\to\infty}\, {\ln n\over b^2_n}= c\in [0,\infty]. \] Then, for any number \(t> 0\), \[ \lim_{n\to\infty}\, {1\over b^2_n}\ln P\Biggl\{{1\over b_n\sqrt{n}}\, \sum^{\alpha_n}_{k=1} \ln{S_k\over k}\geq t\Biggr\}= -\infty, \] where \[ \alpha_n= \begin{cases} b_n\sqrt{n}(\ln n)^{1/10},\quad &\text{if }\lim_{n\to\infty}\, {\ln n\over b^2_n}= c<\infty,\\ \sqrt{n}\ln n,\quad &\text{if }\lim_{n\to\infty}\, {\ln n\over b^2_n}= \infty.\end{cases} \]