Geodesic
Towards Nikishin's Theorem on the Almost Sure Convergence of Rearrangements of Functional Series
Funkcionalʹnyj analiz i ego priloženiâ, Tome 45 (2011) no. 1, pp. 41-55

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Necessary and sufficient conditions are found for the almost sure convergence of almost all simple rearrangements of a series of Banach space valued random variables. The results go back to Nikishin's well-known theorem on the existence of an almost surely convergent rearrangement of a numerical random series. An example is also given of a numerical random series with general term tending to zero almost surely such that this series converges in probability and any its rearrangement diverges almost surely.
Keywords: rearrangement of a series in a Banach space, almost sure convergence, ${\mathbf k}$-simple permutation, Nikishin's theorem. 
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     author = {Sh. Levental and V. S. Mandrekar and S. A. Chobanyan},
     title = {Towards {Nikishin's} {Theorem} on the {Almost} {Sure} {Convergence} of {Rearrangements} of {Functional} {Series}},
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Sh. Levental; V. S. Mandrekar; S. A. Chobanyan. Towards Nikishin's Theorem on the Almost Sure Convergence of Rearrangements of Functional Series. Funkcionalʹnyj analiz i ego priloženiâ, Tome 45 (2011) no. 1, pp. 41-55. http://geodesic.mathdoc.fr/item/FAA_2011_45_1_a3/