Sharp inequalities involving multiplicative chaos sums

TL;DR

This paper extends previous work on multiplicative systems of random variables by establishing a general extreme inequality and demonstrating convergence properties of systems generated by bounded products.

Contribution

It introduces a new extreme inequality for multiplicative systems and shows that systems generated by bounded products are convergence systems.

Findings

Established a general extreme inequality for multiplicative systems.

Proved convergence of systems generated by bounded products.

Derived corollaries for Rademacher chaos sums.

Abstract

The present note is an essential addition to the author's arxiv paper arXiv:2001.01070, concerning general multiplicative systems of random variables. Using some lemmas and the methodology of \cite{Kar4}, we obtain a general extreme inequality, with corollaries involving Rademacher chaos sums and those analogues for multiplicative systems. In particular we prove that a system of functions generated by bounded products of a multiplicative system is a convergence system.