{
  "id": "2303.09488",
  "version": "v1",
  "published": "2023-03-16T17:07:47.000Z",
  "updated": "2023-03-16T17:07:47.000Z",
  "title": "Regularity of laws via Dirichlet forms -- Application to quadratic forms in independent and identically distributed random variables",
  "authors": [
    "Ronan Herry",
    "Dominique Malicet",
    "Guillaume Poly"
  ],
  "comment": "33 pages, comments are welcome",
  "categories": [
    "math.PR",
    "math.FA"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-16T17:07:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H07",
      "31C25",
      "60E05"
    ],
    "keywords": [
      "identically distributed random variables",
      "dirichlet forms",
      "regularity",
      "quadratic forms",
      "independent"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}