{
  "id": "1805.01720",
  "version": "v1",
  "published": "2018-05-04T11:45:23.000Z",
  "updated": "2018-05-04T11:45:23.000Z",
  "title": "Regularity of solutions of the Stein equation and rates in the multivariate central limit theorem",
  "authors": [
    "Thomas Gallouët",
    "G Mijoule",
    "Y Swan"
  ],
  "categories": [
    "math.PR",
    "math.AP",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "Consider the multivariate Stein equation $\\Delta f - x\\cdot \\nabla f = h(x) - E h(Z)$, where $Z$ is a standard $d$-dimensional Gaussian random vector, and let $f\\_h$ be the solution given by Barbour's generator approach. We prove that, when $h$ is $\\alpha$-H\\\"older ($0<\\alpha\\leq1$), all derivatives of order $2$ of $f\\_h$ are $\\alpha$-H\\\"older {\\it up to a $\\log$ factor}; in particular they are $\\beta$-H\\\"older for all $\\beta \\in (0, \\alpha)$, hereby improving existing regularity results on the solution of the multivariate Gaussian Stein equation. For $\\alpha=1$, the regularity we obtain is optimal, as shown by an example given by Rai\\v{c} \\cite{raivc2004multivariate}. As an application, we prove a near-optimal Berry-Esseen bound of the order $\\log n/\\sqrt n$ in the classical multivariate CLT in $1$-Wasserstein distance, as long as the underlying random variables have finite moment of order $3$. When only a finite moment of order $2+\\delta$ is assumed ($0<\\delta<1$), we obtain the optimal rate in $\\mathcal O(n^{-\\frac{\\delta}{2}})$. All constants are explicit and their dependence on the dimension $d$ is studied when $d$ is large.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-04T11:45:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "multivariate central limit theorem",
      "regularity",
      "dimensional gaussian random vector",
      "finite moment",
      "multivariate gaussian stein equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}