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

Abstract

A theorem of Komlοs is a subsequence version of the strong law of large numbers. It states that if (fn)n is a sequence of norm-bounded random variables in L1 (µ), where µ is a probability measure, then there exists a subsequence (gk)k of (fn)m and f ∈ L1(µ) such that for all further subsequences (hm)m, the sequence of successive arithmetic means of {hm)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 Komlοs9 theorem, then C must be L1 -norm bounded. © 1993 by Pacific Journal of Mathematics.