Strong limit theorems for blockwise \(m\)-dependent random variables and generalization of the conjectures of Móricz
type: Item
instance of: scholarly article
title: Strong limit theorems for blockwise \(m\)-dependent random variables and generalization of the conjectures of Móricz
zbMATH Open document ID: 0742.60032
published in: Chinese Annals of Mathematics. Series B
publication date: 1992-06-25T00:00:00Z
review text: Let \(B\) be a real separable Banach space, \(\{X_ n:\;n\geq 1\}\) a sequence of \(B\)-valued r.v.s, \(m\) a nonnegative integer, and \(\{a_ n:\;n\geq 1\}\) a sequence of positive real numbers such that \(a_ 1\geq 1\), \(a_{n+1}-a_ n\geq 1\) for \(n\geq 1\) and \((a_{n+1}-a_ n)\uparrow\infty\). \(\{X_ n\}\) is said to be blockwise \(m\)-dependent with respect to \(\{a_ n\}\) if for large enough \(n\), the two sets \(\{X_{[a_ n]+1},\dots,X_ k\}\) and \(\{X_ \ell,X_{\ell+1},\dots,X_{[a_{n+1}]}\}\) are independent provided \(a_ n<k\), \(\ell\leq a_{n+1}\), and \(\ell-k>m\). In the real-valued case, the reviewer introduced [Proc. Am. Math. Soc. 101, 709-715 (1987; Zbl 0632.60025)] another notion of blockwise \(m\)- dependence, which is a particular case of the above one when \(a_ n=2^ n\) for \(n\geq 1\). The authors prove strong laws of large numbers and the law of the iterated logarithm for such r.v.s. They also give positive answers to the conjectures raised by the reviewer.
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 11938
zbMATH Keywords: separable Banach space
zbMATH Keywords: blockwise \(m\)-dependent
zbMATH Keywords: strong laws of large numbers
zbMATH Keywords: law of the iterated logarithm
author: Deli Li
author: Xiang Chen Wang
reviewed by: Ferenc Móricz
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 11938