Regularization along central convergence on second and third Wiener chaoses

TL;DR

This paper investigates regularization phenomena along central convergence in second and third Wiener chaoses, showing that convergence to Gaussian distributions enhances the regularity of the involved random variables.

Contribution

It extends regularization results to the third Wiener chaos and removes previous assumptions, introducing new techniques involving spectral analysis of Gaussian matrices.

Findings

Regularization along central convergence in second Wiener chaos.

Enhanced non-degeneracy estimates in third Wiener chaos.

Spectral analysis approach for Malliavin derivatives.

Abstract

Consider $F$ an element of the second Wiener chaos with variance one. In full generality, we show that, for every integer $p\ge 1$, there exists $\eta_p>0$ such that if $\kappa_4(F)<\eta_p$ then the Malliavin derivative of $F$ admits a negative moment of order $p$. This entails that any sequence of random variables in the second Wiener chaos converging in distribution to a non--degenerated Gaussian is getting more regular as its distribution is getting close to the normal law. This substantially generalizes some recent findings contained in \cite{hu2014convergence,hu2015density,nourdin2016fisher} where analogous statements were given with additional assumptions which we are able to remove here. Moreover, we provide a multivariate version of this Theorem. Our main contribution concerns the case of the third Wiener chaos which is notoriously more delicate as one cannot anymore decompose the random variables into a linear combination of i.i.d. random variables. We still prove that the same phenomenon of regularization along central convergence occurs. Unfortunately, we are not able to provide a statement as strong as the previous one, but we can show that the usual non--degeneracy estimates of the Malliavin derivative given by the Carbery-Wright inequality can be improved by a factor three. Our proof introduces new techniques such that a specific Malliavin gradient enabling us to encode the distribution of the Malliavin derivative by the spectrum of some Gaussian matrix. This allows us to revisit the fourth moment phenomenon in terms of the behavior of its spectral radius.