{
  "id": "2304.11459",
  "version": "v1",
  "published": "2023-04-22T18:13:54.000Z",
  "updated": "2023-04-22T18:13:54.000Z",
  "title": "Variation comparison between infinitely divisible distributions and the normal distribution",
  "authors": [
    "Ping Sun",
    "Ze-Chun Hu",
    "Wei Sun"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $X$ be a continuous random variable with finite second moment. We investigate the inequality: $P\\{|X-E[X]|\\le \\sqrt{{\\rm Var}(X)}\\}\\ge P\\{|Z|\\le 1\\}$, where $Z$ is a standard normal random variable. We prove that this inequality holds for many familiar infinitely divisible distributions including the log-normal distribution, student's $t$-distribution and the inverse Gaussian distribution. Numerical results are given to show that the inequality with continuity correction also holds for some infinitely divisible discrete distributions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-22T18:13:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E15",
      "62G32",
      "90C15"
    ],
    "keywords": [
      "infinitely divisible distributions",
      "normal distribution",
      "variation comparison",
      "finite second moment",
      "inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}