Journal of Convex Analysis 28 (2021), No. 1, 197--202
Copyright Heldermann Verlag 2021
Permutation-Invariance in Komlós' Theorem for Hilbert-Space Valued Random Variables
Abdessamad Dehaj
Laboratory of Algebra, Analysis and Applications, Department of Mathematics and Computer Science, Faculty of Sciences Ben M'Sik, Hassan II University, Sidi Othman -- Casablanca, Morocco
a.dehaj@gmail.com
Mohamed Guessous
Laboratory of Algebra, Analysis and Applications, Department of Mathematics and Computer Science, Faculty of Sciences Ben M'Sik, Hassan II University, Sidi Othman -- Casablanca, Morocco
guessousjssous@yahoo.fr
[Abstract-pdf]
The Koml\'{o}s theorem states that we can extract a subsequence from every $L_{\mathbb{R}}^{1}$-bounded sequence of random variables, so that every further subsequence converges Ces\`{a}ro a.e. to the same limit. The purpose of this paper is to prove that if $\mathbb{H}$ is a Hilbert space, we can extract a subsequence from every $L_{\mathbb{H}}^{1}$-bounded sequence, so that every permuted subsequence converges Ces\`{a}ro a.e. in $\mathbb{H}$ to the same limit.
Keywords: Bounded sequences, Cesaro-convergence, Hilbert space, Komlos theorem, permutation.
MSC: 28A20, 46B20.
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