Regularity of laws via Dirichlet forms: application to quadratic forms in independent and identically distributed random variables

• 成果类型: Article
• 署名作者: Herry, Ronan; Malicet, Dominique; Poly, Guillaume
• 署名单位: Universite de Rennes; Universite Paris-Est-Creteil-Val-de-Marne (UPEC); Universite Gustave-Eiffel
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-024-01332-x
• 发表日期: 2025
• 页码: 523-567
• 关键词: convergence; POLYNOMIALS; Invariance; densities; THEOREM

摘要

We study the regularity of the law of a quadratic form Q(X, X), evaluated in a sequence X = ( Xi) of independent and identically distributed random variables, when X1 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 = (Xi) a sequence of independent and identically distributed centered random variables, with unit variance, following such distribution. Consider also ( Qn) a sequence of quadratic forms, with associated symmetric Hilbert-Schmidt operators (A(n)). Assume that Tr[(A(n))2] = 1/2, A(n) ii = 0, and the spectral radius of A(n) tends to 0. Then, ( Qn( 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 R to the corresponding derivatives of the standard Gaussian density. While classical methods, from Malliavin calculus or calculus, generally consist in bounding negative moments of the so-called carre du champ operator ( Q( X), Q( X)), we provide a new paradigm through a second-order criterion involving the eigenvalues of aHessian-type matrix related to Q( X). ThisHessian is built by iterating twice a tailor-made gradient, the sharp operator , obtained via a Gaussian representation of the carre du champ. We believe that this method, recently developed by the authors in the current paper and Herry et al. (Ann Probab 52(3):1162-1200, 2024), is of independent interest and could prove useful in other settings.