Two-sided moment estimates for a class of nonnegative chaoses

TL;DR

This paper establishes precise two-sided bounds for the moments of nonnegative random chaoses, which are multilinear forms generated by independent nonnegative variables satisfying a specific growth condition on their moments.

Contribution

The authors provide deterministic, exact-up-to-constants bounds for moments of nonnegative chaoses under a new growth condition on the variables' moments.

Findings

Derived two-sided moment bounds for nonnegative chaoses

Bounds depend only on chaos order and moment growth constant

Results are deterministic and tight up to multiplicative constants

Abstract

We derive two-sided bounds for moments of random multilinear forms (random chaoses) with nonnegative coeficients generated by independent nonnegative random variables $X_i$ which satisfy the following condition on the growth of moments: $\lv X_i \rv_{2p} \leq A \lv X_i \rv_p$ for any $i$ and $p\geq 1$. Estimates are deterministic and exact up to multiplicative constants which depend only on the order of chaos and the constant $A$ in the moment assumption.