Central limit theorem and increment conditions
type: Item
instance of: scholarly article
title: Central limit theorem and increment conditions
zbMATH Open document ID: 0621.60024
DOI: 10.1016/0167-7152(86)90065-9
author: Wansoo T. Rhee
published in: Statistics \& Probability Letters
publication date: 1986-01-01T00:00:00Z
review text: Suppose that \((X_ t,0\leq t\leq 1)\) is a uniformly bounded separable process. A remarkable recent result of \textit{E. Giné} and \textit{J. Zinn} [Ann. Probab. 12, 929-989 (1984; Zbl 0553.60037); Theorem 7.4] shows that if for every \(u<t\), \(| X_ t-X_ n| \leq t-u\), then \((X_ t)\) satisfies the central limit theorem. In the present paper it is shown that if \(\phi\) is a convex function defined on \(R^+\) with \(\phi (0)=0\) and \(\lim_{x\to 0}\phi (x)/x=0\), then there exists a uniformly bounded separable process \((X_ t,0\leq t\leq 1)\) with continuous sample paths such that for every \(u<t\) \(E[\phi (| X_ t-X_ u|)]\leq t-u\) but for which the central limit theorem does not hold. Thus, the Giné and Zinn result is close to optimal.
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 4007336
zbMATH Keywords: increment conditions
zbMATH Keywords: central limit theorem
zbMATH Keywords: continuous sample paths
MaRDI profile type: Publication
full work available at URL: 0167-7152(86)90065-9
OpenAlex ID: W1984063808
cites work: Metric entropy and the central limit theorem in C(S)
cites work: Some limit theorems for empirical processes (with discussion)
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cites work: Q5592735
cites work: Q3903762
description: scientific article; zbMATH DE number 4007336