A converse to a theorem of Komlós for convex subsets of \(L_ 1\)
type: Item
instance of: scholarly article
title: A converse to a theorem of Komlós for convex subsets of \(L_ 1\)
zbMATH Open document ID: 0739.46017
author: Chris Lennard
published in: Pacific Journal of Mathematics
publication date: 1993-01-01T00:00:00Z
review text: A theorem of Komlós is a subsequence version of the strong law of large numbers. It states that if \((f_ n)_ n\) is a sequence of norm-bounded random variables in \(L_ 1(\mu)\), where \(\mu\) is a probability measure, then there exists a subsequence \((g_ k)_ k\) of \(f_ n)_ n\) and \(f\in L_ 1(\mu)\) such that for all further subsequences \((h_ m)_ m\), the sequence of successive arithmetic means of \((h_ m)_ m\) converges to \(f\) almost everywhere. In this paper we show that, conversely, if \(C\) is a convex subset of \(L_ 1(\mu)\) satisfying the conclusion of Komlós theorem, then \(C\) must be \(L_ 1\)-norm bounded.
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 60F25
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 4216245
zbMATH Keywords: subsequence version of the strong law of large numbers
zbMATH Keywords: sequence of norm- bounded random variables
zbMATH Keywords: successive arithmetic means
zbMATH Keywords: Komlós theorem
MaRDI profile type: Publication
full work available at URL: pjm.1993.159.75
OpenAlex ID: W2092173293
DOI: 10.2140/PJM.1993.159.75
description: scientific article; zbMATH DE number 4216245