{
  "id": "1310.3785",
  "version": "v2",
  "published": "2013-10-14T18:51:47.000Z",
  "updated": "2015-11-12T09:12:17.000Z",
  "title": "Characterization of the convergence in total variation and extension of the Fourth Moment Theorem to invariant measures of diffusions",
  "authors": [
    "Seiichiro Kusuoka",
    "Ciprian Tudor"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1109.5381 by other authors",
  "categories": [
    "math.PR"
  ],
  "abstract": "We give necessary and sufficient conditions to characterize the convergence in distribution of a sequence of arbitrary random variables to a probability distribution which is the invariant measure of a diffusion process. This class of target distributions includes the most known continuous probability distributions. Precisely speaking, we characterize the convergence in total variation to target distributions which are not Gaussian or Gamma distributed, in terms of the Malliavin calculus and of the coefficients of the associated diffusion process. We also prove that, among the distributions whose associated squared diffusion coefficient is a polynomial of second degree (with some restrictions on its coefficients), the only possible limits of sequences of multiple integrals are the Gaussian and the Gamma laws.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-10-14T18:51:47.000Z",
      "title": "Extension of the Fourth Moment Theorem to invariant measures of diffusions",
      "abstract": "We characterize the convergence in distribution of a sequence of random variables in a Wiener chaos of a fixed order to a probability distribution which is the invariant measure of a diffusion process. This class of target distributions includes the most known continuous probability distributions. Our results are given in terms of the Malliavin calculus and of the coefficients of the associated diffusion process and extend the standard Fourth Moment Theorem by Nualart and Peccati. In particular we prove that, among the distributions whose associated squared diffusion coefficient is a polynomial of second degree (with some restrictions on its coefficients), the only possible limits of sequences of multiple integrals are the Gaussian and the Gamma laws.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-11-12T09:12:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "invariant measure",
      "standard fourth moment theorem",
      "coefficient",
      "target distributions",
      "continuous probability distributions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.3785K"
    }
  }
}