The celebrated theorem of Komlos (1967) establishes L^1-boundedness as a sufficient condition for a sequence of measurable functions on a probability space to contain a subsequence along which, and along whose every further subsequence (“hereditarily”), the Cesaro averages converge to a “randomized mean” in the spirit of the Strong law of Large Numbers. We provide conditions not only sufficient, but also necessary, for this result, as well as for the hereditary analogues of the Weak Law of Large Numbers, of the Hsu-Robbins-Erdos Law of Large Numbers, and of the Law of the Iterated Logarithm.

Joint work with I. Berkes (Budapest) and W. Schachermayer (Vienna).