Squared chaotic random variables: New moment inequalities with applications

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 generalization, as well as a new analytical proof, of an estimate by Frenkel (2007) [10], and also constitutes a natural real counterpart to an inequality established by Arias-de-Reyna (1998) [2] 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) [19], as well as to obtain a novel probabilistic representation of the remainder in Hadamard inequality of matrix analysis.