A subsequence principle in probability theory
type: Item
instance of: scholarly article
title: A subsequence principle in probability theory
zbMATH Open document ID: 0571.60028
published in: Jahresbericht der Deutschen Mathematiker-Vereinigung (DMV)
publication date: 1985-01-01T00:00:00Z
review text: A survey paper by an author who contributed a large part of the theory of the so-called subsequence principle. Let (\(\Omega\),\(\Sigma\),\(\mu)\) be a measure space and \(g_ n\), \(n\in {\mathbb{N}}\), a sequence of measurable functions \(\Omega\) \(\to {\mathbb{R}}\). Under certain conditions there is a subsequence of \(\{g_ n\}\) behaving as a sequence of exchangeable or even i.i.d. random variables. E.g. if \(\int | g_ n| dP\leq M\) there is a subsequence \(\{f_ n\}\) of \(\{g_ n\}\) such that any subsequence of \(\{f_ n\}\) satisfies the strong law of large numbers. If \(\int | g_ n|^ 2dP\leq M\), a similar result holds with central limit theorem and log log law. The development of the theory is described, especially the fundamental results of \textit{D. J. Aldous}, see the foregoing review, Zbl 0571.60027. The connection with gap theorems and results on functions \(g_ n(x)=f(\lambda_ nx)\), \(\Omega ={\mathbb{R}}\), \(\lambda_{n+1}/\lambda_ n\geq q>1\), is described. The state of the problem for strongly measurable \(g_ n:\Omega \to E\) a Banach space, is reviewed.
Mathematics Subject Classification ID: 60Fxx
Mathematics Subject Classification ID: 60B12
zbMATH DE Number: 3911383
zbMATH Keywords: subsequence principle
zbMATH Keywords: exchangeable
zbMATH Keywords: strong law of large numbers
zbMATH Keywords: central limit theorem
author: Srishti D. Chatterji
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3911383