On the almost sure convergence of sums

Luca Pratelli and Pietro Rigo

Statistics & Probability Letters, 2021, vol. 172, issue C

Abstract: Two counterexamples, addressing questions raised in Adamczak (2019) and Poly and Zheng (2019), are provided. Both counterexamples are related to chaoses. Let Fn=Yn+Zn, where the random variables Yn and Zn belong to different chaoses of uniformly bounded degree. It may be that Fn⟶a.s.0, Fn⟶L2+δ0 and E{supn|Fn|δ}<∞, for some δ>0, and yet Yn fails to converge to 0 a.s.

Keywords: Almost sure convergence; Homogeneous sums; Poisson chaos (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1016/j.spl.2021.109045

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