Regularity of laws via Dirichlet forms -- Application to quadratic forms in independent and identically distributed random variables

Abstract

We study the regularity of the law of a quadratic form $Q(X,X)$, evaluated in a sequence $X = (X_{i})$ of independent and identically distributed random variables, when $X_{1}$ can be expressed as a sufficiently smooth function of a Gaussian field. This setting encompasses a large class of important and frequently used distributions, such as, among others, Gaussian, Beta, for instance uniform, Gamma distributions, or else any polynomial transform of them. Let us present an emblematic application. Take $X = (X_{i})$ a sequence of independent and identically distributed centered random variables, with unit variance, following such distribution. Consider also $(Q_{n})$ a sequence of quadratic forms, with associated symmetric Hilbert--Schmidt operators $(\mathsf{A}^{(n)})$. Assume that $\operatorname{Tr}[ (\mathsf{A}^{(n)})^{2} ] = 1/2$, $\mathsf{A}^{(n)}_{ii} =0$, and the spectral radius of $\mathsf{A}^{(n)}$ tends to $0$. Then, $(Q_{n}(X))$ converges in a strong sense to the standard Gaussian distribution. Namely, all derivatives of the densities, which are well-defined for $n$ sufficiently large, converge uniformly on $\mathbb{R}$ to the corresponding derivatives of the standard Gaussian density. While classical methods, from Malliavin calculus or $\Gamma$-calculus, generally consist in bounding negative moments of the so-called \emph{carr\'e du champ} operator $\Gamma(Q(X),Q(X))$, we provide a new paradigm through a second-order criterion involving the eigenvalues of a Hessian-type matrix related to $Q(X)$. This Hessian is built by iterating twice a tailor-made gradient, the \emph{sharp operator} $\sharp$, obtained via a Gaussian representation of the carr\'e du champ. We believe that this method, recently developed by the authors in the current paper and in their companion paper [AoP 52 n{\deg}3 (2024)] , is of independent interest and could prove useful in other settings.