An \(L_ 1\)-convergence theorem for heterogeneous mixingale arrays with trending moments
type: Item
instance of: scholarly article
title: An \(L_ 1\)-convergence theorem for heterogeneous mixingale arrays with trending moments
zbMATH Open document ID: 0783.60024
DOI: 10.1016/0167-7152(93)90134-5
author: S. Singh
published in: Statistics \& Probability Letters
publication date: 1993-05-16T00:00:00Z
review text: Consider an array of pairs \(\bigl\{ X_{nt},F_{nt}:-\infty<t<\infty\), \(n \geq 1 \bigr\}\) where \(X_{nt}\) are random variables defined on a probability space \((\Omega,F,P)\) and \(F_{nt}\) are \(\sigma\)-subfields of \(F\), increasing in \(t\). The array is an \({\mathcal L}_ 1\)-mixingale in the sense that there exist constants \(\{c_{nt}\}\) and a non-negative sequence \(\{\zeta_ m\}\) converging to zero when \(n \to \infty\) such that \(\bigl\| {\mathcal E}(X_{nt} \mid F_{n,t-m}) \bigr\|_ 1\leq c_{nt}\zeta_ m\) and \(\bigl\| X_{nt}-{\mathcal E}(X_{nt}\mid F_{n,t-m}) \bigr\|_ 1 \leq c_{nt}\zeta_{m+1}\) for all \(t\), \(n\) and \(m \geq 0\). For such an array, the following theorem has been proved: Suppose \(\limsup_{n \to \infty} \sum^{k_ n}_{t=1} c_{nt}< \infty\), \(\limsup_{n \to \infty}\sum^{k_ n}_{t=1}c^ 2_{nt}=0\) and \(\{X_{nt}/C_{nt}\}\) is uniformly integrable, \(k_ n\) being an increasing, integer-valued function of \(n\) and \(k_ n \to \infty\) as \(n \to \infty\). Then \(\lim_{n \to \infty}{\mathcal E} \bigl|\sum^{k_ n}_{t=1}X_{nt} \bigr|=0\).
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 168304
zbMATH Keywords: weak law of large numbers
zbMATH Keywords: mixingales
zbMATH Keywords: non-stationarity
reviewed by: Sujit K. Basu
MaRDI profile type: Publication
full work available at URL: 0167-7152(93)90134-5
OpenAlex ID: W2036381517
cites work: On the $L_p$-Convergence for $n^{-1/p} S_n, 0 &lt; p &lt; 2^1$
cites work: Q3911791
cites work: A maximal inequality and dependent strong laws
cites work: Invariance principles for dependent variables
cites work: On the invariance principle for nonstationary mixingales
description: scientific article; zbMATH DE number 168304