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

Ronan Herry, Dominique Malicet, Guillaume Poly

Published 2023-03-16Version 1

We present a new tool to study the regularity of a function $F$ of a sequence $(X_{i})$ of independent and identically distributed random variables. Our main result states that, under mild conditions on the law of $X_{1}$, the regularity of the law of $F$ is controlled by the regularity of the law of a conditionally Gaussian object, canonically associated with $F$. At the technical level our analysis relies on the formalism of Dirichlet forms and an explicit construction of the Malliavin derivative of $F$ in the direction of a Gaussian space. As an application, we derive an explicit control of the regularity of the law of a quadratic from in the $X_{i}$'s in terms of spectral quantities, when the law of $X_{1}$ belongs to a large class of distribution including, for instance, all the Gaussian, all the Beta, all the Gamma, and all the polynomials thereof.