The central limit theorem for empirical and quantile processes in some Banach spaces
type: Item
instance of: scholarly article
title: The central limit theorem for empirical and quantile processes in some Banach spaces
zbMATH Open document ID: 0780.60011
DOI: 10.1016/0304-4149(93)90083-G
published in: Stochastic Processes and their Applications
publication date: 1994-01-30T00:00:00Z
review text: Let \(\alpha_ n(t)\), \(0<t<1\), \(n\in N\), be a sequence of empirical processes constructed by means of a sequence of independent uniformly distributed on \((0,1)\) random variables. The sequence of finite- dimensional distributions of \((\alpha_ n)\) converges weakly to those of Brownian bridge \(B(t)\), \(0\leq t\leq 1\). Let also \(X\) be a Banach function space on \((0,1)\) such that the realizations of \(B\) and \(\alpha_ n\), \(n\in N\), belong to \(X\) a.s. Then, for which \(X\) we can say that the sequence of distributions of \(\alpha_ n\), \(n\in N\), in \(X\) converges weakly to the distribution of \(B\) in \(X\)? This question is answered affirmatively when \(X\) is the Orlicz space \(L_ \Phi\), where \(\Phi\) satisfies the \(\Delta_ 2\)-condition. This is derived from obvious results concerning the CLT in Banach spaces.
Mathematics Subject Classification ID: 60B12
Mathematics Subject Classification ID: 62G30
zbMATH DE Number: 203255
zbMATH Keywords: \(p\)-convexity
zbMATH Keywords: \(q\)-concavity
zbMATH Keywords: empirical processes
zbMATH Keywords: Brownian bridge
zbMATH Keywords: Orlicz space
zbMATH Keywords: central limit theorem in Banach spaces
author: Rimas Norvaiša
reviewed by: V. Tarieladze
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 203255