Some applications of the Menshov-Rademacher theorem

2021/03/17 by Mukeru, Safari
#FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR)

paper · doi:10.48550/arxiv.2103.09594

Abstract

Given a sequence (X_n) of real or complex random variables and a sequence of numbers (a_n), an interesting problem is to determine the conditions under which the series ∑_n=1^∞ a_n X_n is almost surely convergent. This paper extends the classical Menshov--Rademacher theorem on the convergence of orthogonal series to general series of dependent random variables and derives interesting sufficient conditions for the almost everywhere convergence of trigonometric series with respect to singular measures whose Fourier transform decays to 0 at infinity with positive rate.

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