A converse to a theorem of Komlós for convex subsets of L₁

Christopher John Lennard

Vol. 159 (1993), No. 1, 75–85

DOI: 10.2140/pjm.1993.159.75

Abstract

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 L₁(μ), where μ is a probability measure, then there exists a subsequence (g_k)_k of (f_n)_n and f ∈ L₁(μ) 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 L₁(μ) satisfying the conclusion of Komlós’ theorem, then C must be L₁-norm bounded.

Mathematical Subject Classification 2000

Primary: 28A20

Secondary: 60F15

Milestones

Received: 3 October 1991

Published: 1 May 1993