{
  "id": "1912.04605",
  "version": "v1",
  "published": "2019-12-10T10:01:58.000Z",
  "updated": "2019-12-10T10:01:58.000Z",
  "title": "On algebraic Stein operators for Gaussian polynomials",
  "authors": [
    "Ehsan Azmoodeh",
    "Dario Gasbarra",
    "Robert E. Gaunt"
  ],
  "comment": "60 pages. This arXiv version has an extended Appendix B of Stein operators for univariate Gaussian polynomials compared to the version submitted for publication",
  "categories": [
    "math.PR"
  ],
  "abstract": "The first essential ingredient to build up Stein's method for a continuous target distribution is to identify a so-called \\textit{Stein operator}, namely a linear differential operator with polynomial coefficients. In this paper, we introduce the notion of \\textit{algebraic} Stein operators (see Definition \\ref{def:algebraic-Stein-Operator}), and provide a novel algebraic method to find \\emph{all} the algebraic Stein operators up to a given order and polynomial degree for a target random variable of the form $Y=h(X)$, where $X=(X_1,\\dots, X_d)$ has i.i.d$.$ standard Gaussian components and $h\\in \\KK[X]$ is a polynomial with coefficients in the ring $\\KK$. Our approach links the existence of an algebraic Stein operator with \\textit{null controllability} of a certain linear discrete system. A \\texttt{MATLAB} code checks the null controllability up to a given finite time $T$ (the order of the differential operator), and provides all \\textit{null control} sequences (polynomial coefficients of the differential operator) up to a given maximum degree $m$. This is the first paper that connects Stein's method with computational algebra to find Stein operators for highly complex probability distributions, such as $H_{20}(X_1)$, where $H_p$ is the $p$-th Hermite polynomial. A number of examples of Stein operators for $H_p(X_1)$, $p=3,4,5,6,7,8,10,12$, are gathered in the extended Appendix of this arXiv version. We also introduce a widely applicable approach to proving that Stein operators characterise the target distribution, and use it to prove, amongst other examples, that the Stein operators for $H_p(X_1)$, $p=3,\\ldots,8$, with minimum possible maximal polynomial degree $m$ characterise their target distribution.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-10T10:01:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "algebraic stein operator",
      "gaussian polynomials",
      "target distribution",
      "polynomial coefficients",
      "steins method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 60,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}