New proofs of a theorem of Komlós (Q1088283)

Given an \(L^ 1\)-bounded sequence of random variables \(\{X_ n\}\) on a probability space (\(\Omega\),\({\mathcal F},P)\), there exists a subsequence \(\{Y_ n\}\) such that every further subsequence converges Cesaro a.s. to an integrable random variable X. The limit X is independent of the particular subsequence of \(\{Y_ n\}.\) The original proof of this result, due to \textit{J. Komlós} [Acta Math. Acad. Sci. Hung. 18, 217-229 (1967; Zbl 0228.60012)], uses martingale difference sequences. In this paper, new proofs of Komlós' theorem are given. An argument based on a maximal inequality and an approach patterned after a proof by \textit{N. Etemadi} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 55, 119-122 (1981; Zbl 0438.60027)], of the strong law of large numbers are given.