{
  "id": "2310.03471",
  "version": "v1",
  "published": "2023-10-05T11:27:45.000Z",
  "updated": "2023-10-05T11:27:45.000Z",
  "title": "On the measure concentration of infinitely divisible distributions",
  "authors": [
    "Jing Zhang",
    "Ze-Chun Hu",
    "Wei Sun"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let ${\\cal I}$ be the set of all infinitely divisible random variables\\ with finite second moments, ${\\cal I}_0=\\{X\\in{\\cal I}:{\\rm Var}(X)>0\\}$, $P_{\\cal I}=\\inf_{X\\in{\\cal I}}P\\{|X-E[X]|\\le \\sqrt{{\\rm Var}(X)}\\}$ and $P_{{\\cal I}_0}=\\inf_{X\\in{\\cal I}_0} P\\{|X-E[X]|< \\sqrt{{\\rm Var}(X)}\\}$. We prove that $P_{{\\cal I}}\\ge P_{{\\cal I}_0}>0$. Further, we use geometric and Poisson distributions to investigate the values of $P_{\\cal I}$ and $P_{{\\cal I}_0}$. In particular, we show that $P_{\\cal I}\\le e^{-1}\\sum_{k=0}^{\\infty}\\frac{1}{2^{2k}(k!)^2}\\approx 0.46576$ and $P_{{\\cal I}_0}\\le e^{-1}\\approx 0.36788$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-05T11:27:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E07",
      "60E15",
      "62G32"
    ],
    "keywords": [
      "infinitely divisible distributions",
      "measure concentration",
      "finite second moments",
      "poisson distributions",
      "infinitely divisible random"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}