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

With D. Malicet & G. Poly. .

We present a new tool to study the regularity of a function F of a sequence (Xi) of independent and identically distributed random variables. Our main result states that, under mild conditions on the law of X1, 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 Xi’s in terms of spectral quantities, when the law of X1 belongs to a large class of distribution including, for instance, all the Gaussian, all the Beta, all the Gamma, and all the polynomials thereof.