Permutation-Invariance in Komlós' Theorem for Hilbert-Space Valued Random Variables

Abdessamad Dehaj; Mohamed Guessous

doi:10.68381/jca28014

Abstract

The Komlós 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 Cesàro 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 Cesàro a.e. in

\mathbb{H}

to the same limit.

Author information

Contact details are reproduced from the original publication and may be historical.

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

Keywords

Bounded sequences
Cesàro-convergence
Hilbert space
Komlós theorem
permutation

Mathematics Subject Classification

28A20, 46B20

References

D. J. Aldous: Limit theorems for subsequences of arbitrarily-dependent sequences of random variables, Prob. Theory Rel. Fields 40(1) (1977) 59–82.https://doi.org/10.1007/BF00535707 Google Scholar
I. Berkes: An extension of the Komlós subsequence theorem, Acta Math. Hungarica 55(1-2) (1990) 103–110.https://doi.org/10.1007/BF01951392 Google Scholar
S. D. Chatterji: Un principe de sous-suites dans les espaces de Banach, in: Séminaire Probabilités XIII, Lecture Notes in Mathematics 721, Springer, Berlin (1972) 4–21.https://doi.org/10.1007/BFb0070848 Google Scholar
N. Etemadi: An elementary proof of the strong law of large numbers, Zeitschrift f. Wahrscheinlichkeitstheorie Verw. Gebiete 55(1) (1981) 119–122.https://doi.org/10.1007/BF01013465 Google Scholar
M. Guessous: An elementary proof of Komlós-Revesz Theorem in Hilbert spaces, J. Convex Analysis 4(2) (1997) 321–332.https://doi.org/10.68381/jca04017 Google Scholar
J. Komlós: A generalization of a problem of Steinhaus, Acta Math. Hungarica 18(1-2) (1967) 217–229.https://doi.org/10.1007/BF02020976 Google Scholar
J. Komlós: Every sequence converging to 0 weakly in L2 contains an unconditional convergence sequence, Arkiv f. Matematik 12(1-2) (1974) 41–49.https://doi.org/10.1007/BF02384745 Google Scholar
A. Meyerowitz, M. Schwartz: Dominated permutations of subsequences of random variables, Trans. Amer. Math. Soc. 294(1) (1986) 287–294.https://doi.org/10.1090/S0002-9947-1986-0819948-6 Google Scholar
M. Schwartz: New proofs of a theorem of Komlós, Acta Math. Hungarica 47(1-2) (1986) 181–185.https://doi.org/10.1007/BF01949141 Google Scholar

Suggested citation

Published by Heldermann Verlag, 2021. Rights now held by Banach Press.

[1] D. J. Aldous: Limit theorems for subsequences of arbitrarily-dependent sequences of random variables, Prob. Theory Rel. Fields 40(1) (1977) 59–82.https://doi.org/10.1007/BF00535707 Google Scholar

[2] I. Berkes: An extension of the Komlós subsequence theorem, Acta Math. Hungarica 55(1-2) (1990) 103–110.https://doi.org/10.1007/BF01951392 Google Scholar

[3] S. D. Chatterji: Un principe de sous-suites dans les espaces de Banach, in: Séminaire Probabilités XIII, Lecture Notes in Mathematics 721, Springer, Berlin (1972) 4–21.https://doi.org/10.1007/BFb0070848 Google Scholar

[4] N. Etemadi: An elementary proof of the strong law of large numbers, Zeitschrift f. Wahrscheinlichkeitstheorie Verw. Gebiete 55(1) (1981) 119–122.https://doi.org/10.1007/BF01013465 Google Scholar

[5] M. Guessous: An elementary proof of Komlós-Revesz Theorem in Hilbert spaces, J. Convex Analysis 4(2) (1997) 321–332.https://doi.org/10.68381/jca04017 Google Scholar

[6] J. Komlós: A generalization of a problem of Steinhaus, Acta Math. Hungarica 18(1-2) (1967) 217–229.https://doi.org/10.1007/BF02020976 Google Scholar

[7] J. Komlós: Every sequence converging to 0 weakly in L2 contains an unconditional convergence sequence, Arkiv f. Matematik 12(1-2) (1974) 41–49.https://doi.org/10.1007/BF02384745 Google Scholar

[8] A. Meyerowitz, M. Schwartz: Dominated permutations of subsequences of random variables, Trans. Amer. Math. Soc. 294(1) (1986) 287–294.https://doi.org/10.1090/S0002-9947-1986-0819948-6 Google Scholar

[9] M. Schwartz: New proofs of a theorem of Komlós, Acta Math. Hungarica 47(1-2) (1986) 181–185.https://doi.org/10.1007/BF01949141 Google Scholar