Limiting distribution of sums of nonnegative stationary random variables
type: Item
instance of: scholarly article
title: Limiting distribution of sums of nonnegative stationary random variables
zbMATH Open document ID: 0551.60021
DOI: 10.1007/BF02481972
author: Simeon M. Berman
published in: Annals of the Institute of Statistical Mathematics
publication date: 1984-01-01T00:00:00Z
review text: The paper presents a limit theorem on the convergence of the distribution of the sum in an array of nonnegative random variables. The array is stationary in the following sense: if the array is represented as \(\{X_{n,j}\), \(1\leq j\leq n<\infty \}\), then, for each n, the random variables \(X_{n,j}\), \(j=1,...,n\), form a stationary (though finite) sequence. The hypothesis of this theorem includes certain kinds of mixing conditions which provide enough independence to give results similar to those for independent summands. The novelty of this work is the introduction of a new mixing condition having more flexibility in certain applications. The main theorem. Assume the following two conditions: (i) There exists a nonincreasing function H(x), \(x\geq 0\), such that \(n\int^{x}_{0}\min (y,1)dP[X_{n,0}>y]\to \int^{x}_{0}\min (y,1)dH(y)\), in the sense of complete convergence on \(x>0\). (ii) There exists a sequence \(\{\gamma_ n\}\) of positive integers such that for every \(c>0\), \(n\sum_{1\leq j\leq \gamma_ n}P[X_{n,0}>c\), \(X_{n,j}>c]\to 0\) and, for every \(q>0\), \(P[X_{n,0}>x| X_{n,j}\), \(j\leq -q\gamma_ n]/P[X_{n,0}>x]\to^{P}1\) (P - in probability). Then \(X_{n,1}+...+X_{n,n}\) has, for \(n\to \infty\), a limiting distribution with the Laplace-Stieltjes transform \(\phi (s)=\exp [\int^{\infty}_{0}(1-e^{-sx})dH(x)].\) There are also given applications to functions defined on a stationary sequence and to the limiting distribution of the maximum in a stationary sequence.
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G10
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 3878060
zbMATH Keywords: mixing conditions
zbMATH Keywords: distribution of the maximum in a stationary sequence
reviewed by: Dominik Szynal
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02481972
OpenAlex ID: W2047924197
description: scientific article; zbMATH DE number 3878060