Squared chaotic random variables: new moment inequalities with applications

TL;DR

This paper introduces new inequalities for squared Gaussian chaos variables, generalizing previous results, and applies them to derive bounds on polynomials and matrix inequalities, advancing theoretical understanding in Gaussian analysis.

Contribution

It provides a new family of inequalities for squared Wiener chaos variables, offering a novel analytical proof and extending prior estimates in Gaussian analysis.

Findings

New inequalities for squared Wiener chaos variables

Improved lower bounds on homogeneous polynomials

Probabilistic representation of Hadamard inequality remainder

Abstract

We prove a new family of inequalities involving squares of random variables belonging to the Wiener chaos associated with a given Gaussian field. Our result provides a substantial generalisation, as well as a new analytical proof, of an estimate by Frenkel (2007), and also constitute a natural real counterpart to an inequality established by Arias-de-Reyna (1998) in the framework of complex Gaussian vectors. We further show that our estimates can be used to deduce new lower bounds on homogeneous polynomials, thus partially improving results by Pinasco (2012), as well as to obtain a novel probabilistic representation of the remainder in Hadamard inequality of matrix analysis.