Some results on convergence rates for probabilities of moderate deviations for sums of random variables
type: Item
instance of: scholarly article
title: Some results on convergence rates for probabilities of moderate deviations for sums of random variables
zbMATH Open document ID: 0753.60028
DOI: 10.1155/S0161171292000644
author: M. Bhaskara Rao
author: Deli Li
author: Xiang Chen Wang
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1993-01-16T00:00:00Z
full work available at URL: https://eudml.org/doc/46731
review text: Summary: Let \(X\), \(X_ n\), \(n\geq 1\), be a sequence of i.i.d. real random variables, and \(S_ n=\sum^ n_{k=1}X_ k\), \(n\geq 1\). Convergence rates of moderate deviations are derived, i.e., the rate of convergence to zero of certain tail probabilities of the partial sums are determined. For example, we obtain equivalent conditions for the convergence of series \(\sum_{n\geq 1}(\psi^ 2(n)/n)P(| S_ n|\geq\sqrt n\varphi(n))\) only under the assumptions that \(EX=0\) and \(EX^ 2=1\), where \(\varphi\) and \(\psi\) are taken from a broad class of functions. These results generalize and improve some recent results of \textit{D. Li} [Sci. China, Ser. A 34, No. 4, 395--404 (1991; Zbl 0735.60031)] and \textit{M. Gafurov} [Probability theory and mathematical statistics, Proc. 4th USSR Jap. Symp., Tbilisi/USSR 1982, Lect. Notes Math. 1021, 137--144 (1983; Zbl 0533.60039)] and some previous work of \textit{J. A. Davis} [Ann. Math. Stat. 39, 1479--1485 (1968; Zbl 0174.49902)]. For \(b\in[0,1]\) and \(\varepsilon>0\), let \[ \lambda_{\varepsilon,b}=\sum_{n\geq 3}((\log\log n)^ b/n)I(| S_ n|\geq\sqrt{(2+\varepsilon)n\log\log n}). \] The behaviour of \(E\lambda_{\varepsilon,b}\) as \(\varepsilon\downarrow 0\) is also studied.
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60B12
Mathematics Subject Classification ID: 60F99
zbMATH DE Number: 89242
zbMATH Keywords: Convergence rates
zbMATH Keywords: moderate deviations
zbMATH Keywords: tail probabilities
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 89242