On the almost sure convergence of sums

2020/09/07 by Pratelli, Luca, Rigo, Pietro
#60A05 #60F99 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2009.03391

Abstract

Two counterexamples, addressing questions raised in \citeAD and \citePZ, are provided. Both counterexamples are related to chaoses. Let F_n=Y_n+Z_n. It may be that F_n\overseta.s.\longrightarrow 0, F_n\oversetL_2+δ\longrightarrow 0 and E\sup_n \absF_n^δ\0 and Y_n and Z_n belong to chaoses of uniformly bounded degree, and yet Y_n fails to converge to 0 a.s.

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