A converse to a theorem of Komlós for convex subsets of \(L_ 1\)
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.
subsequence version of the strong law of large numbers
sequence of norm- bounded random variables
scientific article; zbMATH DE number 4216245