On almost sure convergence of random variables with finite chaos\n decomposition

2019/09/20 by Radosław Adamczak, Adamczak, Radosław
Economics, Econometrics and Finance · Mathematics · #60B11 #60F99 #60H05 #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1909.09576

openalex publication_date 2019/09/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Under mild conditions on a family of independent random variables (X_n) we\nprove that almost sure convergence of a sequence of tetrahedral polynomial\nchaoses of uniformly bounded degrees in the variables (X_n) implies the\nalmost sure convergence of their homogeneous parts. This generalizes a recent\nresult due to Poly and Zheng obtained under stronger integrability conditions.\nIn particular for i.i.d. sequences we provide a simple necessary and sufficient\ncondition for this property to hold.\n We also discuss similar phenomena for sums of multiple stochastic integrals\nwith respect to Poisson processes, answering a question by Poly and Zheng.\n

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