Tail and moment estimates for a class of random chaoses of order two

TL;DR

This paper establishes precise two-sided bounds for the moments and tail probabilities of quadratic forms generated by independent symmetric random variables satisfying a specific moment growth condition, with results being deterministic and sharp up to constants.

Contribution

It provides new deterministic and sharp bounds for moments and tails of quadratic forms of symmetric random variables under a moment growth condition.

Findings

Derived two-sided bounds for moments and tails of quadratic forms

Results are deterministic and exact up to constants depending on lpha

Applicable to a class of symmetric random variables with controlled moment growth

Abstract

We derive two-sided bounds for moments and tails of random quadratic forms (random chaoses of order $2$), generated by independent symmetric random variables such that $\lVert X \rVert_{2p} \leq \alpha \lVert X \rVert_p$ for any $p\geq 1$ and some $\alpha\geq 1$. Estimates are deterministic and exact up to some multiplicative constants which depend only on $\alpha$.