[PDF] Regularity of laws via Dirichlet forms -- Application to quadratic forms in independent and identically distributed random variables-论文阅读讨论-ReadPaper

摘要

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 mainresult states that, under mild conditions on the law of $X_{1}$, the regularityof the law of $F$ is controlled by the regularity of the law of a conditionallyGaussian object, canonically associated with $F$. At the technical level ouranalysis relies on the formalism of Dirichlet forms and an explicitconstruction of the Malliavin derivative of $F$ in the direction of a Gaussianspace. As an application, we derive an explicit control of the regularity ofthe 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, forinstance, all the Gaussian, all the Beta, all the Gamma, and all thepolynomials thereof.

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