Moderate deviations principle for products of sums of random variables
type: Item
instance of: scholarly article
title: Moderate deviations principle for products of sums of random variables
zbMATH Open document ID: 1229.60033
author: Yu Miao
author: Jianyong Mu
published in: Science China. Mathematics
publication date: 2011-07-01T00:00:00Z
review text: 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 \[ \alpha_n= \begin{cases} b_n\sqrt{n}(\ln n)^{1/10},\quad &\text{if }\lim_{n\to\infty}\, {\ln n\over b^2_n}= c<\infty,\\ \sqrt{n}\ln n,\quad &\text{if }\lim_{n\to\infty}\, {\ln n\over b^2_n}= \infty.\end{cases} \]
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60G05
zbMATH DE Number: 5916822
zbMATH Keywords: moderate deviations principle
zbMATH Keywords: products of sums
zbMATH Keywords: independent identically distribution
zbMATH Keywords: positive random variables
MaRDI profile type: Publication
full work available at URL: s11425-011-4195-8
OpenAlex ID: W1965380523
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DOI: 10.1007/S11425-011-4195-8
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description: scientific article; zbMATH DE number 5916822