A CONVERSE TO A THEOREM OF KOMLOS FOR CONVEX SUBSETS OF L(1)

A theorem of Komlos 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(mu), where mu is a probability measure, then there exists a subsequence (g(k))k of (f(n))n and f is-an-element-of L1(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 L1(mu) satisfying the conclusion of Komlos' theorem, then C must be L1-norm bounded.