Tail and moment estimates for chaoses generated by symmetric random variables with logarithmically concave tails

TL;DR

This paper provides sharp two-sided estimates for the moments and tail probabilities of polynomial chaoses generated by symmetric random variables with log-concave tails, extending to arbitrary order with exponential variables.

Contribution

It introduces optimal bounds for chaos moments and tails involving only deterministic quantities, applicable to various orders and specific distributions.

Findings

Two-sided estimates are optimal up to constants.

Results apply to polynomial chaoses of order up to three and arbitrary order with exponential variables.

Estimates depend solely on deterministic parameters.

Abstract

We present two-sided estimates of moments and tails of polynomial chaoses of order at most three generated by independent symmetric random variables with log-concave tails as well as for chaoses of arbitrary order generated by independent symmetric exponential variables. The estimates involve only deterministic quantities and are optimal up to constants depending only on the order of the chaos variable.