{
  "id": "1907.10050",
  "version": "v1",
  "published": "2019-07-23T17:59:35.000Z",
  "updated": "2019-07-23T17:59:35.000Z",
  "title": "On Stein's Method for Multivariate Self-Decomposable Laws",
  "authors": [
    "Benjamin Arras",
    "Christian Houdré"
  ],
  "comment": "64 pages",
  "categories": [
    "math.PR",
    "math.FA"
  ],
  "abstract": "In this work, we pursue the investigation and the development of Stein's method in the infinitely divisible setting and its relation with functional analysis. Section $2$ starts with standard notations and definitions together with a multidimensional characterization theorem for infinitely divisible distributions with finite first moment. Based on this result and on a truncation procedure, Section $3$ develops characterization results for multivariate self-decomposable laws without finite first moment highlighting the role of the L\\'evy-Khintchine representation of the characteristic function of the target self-decomposable distribution. In particular, these results apply to multivariate stable laws with stability parameter belonging to $(0,1]$. In Section $4$, Stein's equation for self-decomposable distributions without finite first moment is set down and solved thanks to a combination of semigroup technics and Fourier analysis. Finally, in the last section of this note, we take a new look at Poincar\\'e-type inequalities for self-decomposable laws with finite first moment. A proof based on semigroup and on Fourier analysis is presented. Several algebraic quantities from Markov diffusion operators theory are computed in this non-local setting and in particular for the rotationally invariant $\\alpha$-stable laws with $\\alpha \\in (1,2)$. Finally, rigidity and stability results for the Poincar\\'e $U$-functional of the rotationally invariant $\\alpha$-stable distribution are obtained thanks to spectral analysis and Dirichlet form theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-23T17:59:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E07",
      "60E10",
      "60F05"
    ],
    "keywords": [
      "multivariate self-decomposable laws",
      "finite first moment",
      "steins method",
      "markov diffusion operators theory",
      "fourier analysis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 64,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}