Some applications of the Menshov-Rademacher theorem

TL;DR

This paper extends the Menshov-Rademacher theorem to dependent random variables and provides conditions for the almost sure convergence of certain series, including trigonometric series with singular measures.

Contribution

It generalizes classical convergence results to dependent variables and explores convergence under singular measures with decaying Fourier transforms.

Findings

Extended Menshov-Rademacher theorem to dependent variables.

Derived conditions for almost sure convergence of series.

Analyzed convergence of trigonometric series with singular measures.

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 $\sum_{n=1}^\infty a_n X_n$ 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.