Weak convergence of partial sums of absolutely regular sequences
type: Item
instance of: scholarly article
title: Weak convergence of partial sums of absolutely regular sequences
zbMATH Open document ID: 0564.60025
DOI: 10.1016/0167-7152(84)90067-1
author: Ernst Eberlein
published in: Statistics \& Probability Letters
publication date: 1984-01-01T00:00:00Z
review text: Suppose \((X_ 1, X_ 2,...)\) is a stationary process. Define \(\beta(n)\) as \(E\{\sup_{A in B^{k+n+1}}^\infty| P(A| B^ k_ 1)-P(A)| \}\), where \(B^ m_ n\) denotes the \(\sigma\)-algebra generated by the variables \((X_ n,...,X_ m)\). If \(\lim_{n\to \infty}\beta (n)=0\), the process is called absolutely regular. Suppose \((X_ 1, X_ 2,...)\) is a stationary absolutely regular sequence of real-valued random variables with zero mean and \((2+\delta)\)-th absolute moment, where \(\delta >0\). Suppose \(\beta (n)<<n^{-(1+\epsilon)(1+2/\delta)}\) for some \(\epsilon >0\). Let \(S_ n\) denote \(X_ 1+...+X_ n\). Then \(\lim_{n\to \infty}n^{-1}Var(S_ n)\) exists: denote this limit by \(\sigma^ 2\). If \(\sigma^ 2>0\), it is shown that the asymptotic distribution of (n \(\sigma\) \({}^ 2)^{- 1/2}S_ n\) is the standard normal distribution.
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 3899848
zbMATH Keywords: stationary absolutely regular sequence
zbMATH Keywords: asymptotic distribution
reviewed by: Lionel Weiss
MaRDI profile type: Publication
full work available at URL: 0167-7152(84)90067-1
OpenAlex ID: W1970823653
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description: scientific article; zbMATH DE number 3899848