A large deviation principle for weighted sums of independent identically distributed random variables
type: Item
instance of: scholarly article
title: A large deviation principle for weighted sums of independent identically distributed random variables
zbMATH Open document ID: 0967.60025
author: Rüdiger Kiesel
author: Ulrich Stadtmüller
published in: Journal of Mathematical Analysis and Applications
publication date: 2001-08-30T00:00:00Z
review text: Let \(X_0,X_1, \dots\) be i.i.d. random variables such that \(EX_0 =0\), \(EX^2_0=1\) and \(M(t)= E(\exp\{X_0\}) <\infty\), \(|t|< \infty\). Let \(a_j(\lambda)\), \(j=0,1,2, \dots\), \(\lambda >0\), where \(\lambda\) is a continuous parameter or \(\lambda\in N\), be real numbers such that for \(\nu=1\) and hence for each \(\nu=1,2, \dots\), \(\sum^\infty_{j=0} |a_j(\lambda) |^\nu <\infty\) for all \(\lambda\). The authors considers the following weighted means of the sequence \(\{X_n,n\geq 0\}\): \(V(\lambda)= \sum^\infty_{j=0} a_j(\lambda)X_j\). Starting from Cramer's theorem on large deviations of the Cesàro means \((X_0+ X_1+ \cdots+ X_{n-1})/n\), the authors show that other weighted sums \(V(n)\) exhibit a similar large deviation behaviour if the weights satisfy certain regularity conditions.
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 40G05
zbMATH DE Number: 1556952
zbMATH Keywords: large deviations
zbMATH Keywords: weighted sums
zbMATH Keywords: independent identically distributed random variables
zbMATH Keywords: Cesàro means
reviewed by: Zdzisław Rychlik
MaRDI profile type: Publication
full work available at URL: jmaa.2000.7176
OpenAlex ID: W1981301100
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DOI: 10.1006/JMAA.2000.7176
description: scientific article; zbMATH DE number 1556952