{
  "id": "2207.01872",
  "version": "v1",
  "published": "2022-07-05T08:13:35.000Z",
  "updated": "2022-07-05T08:13:35.000Z",
  "title": "Optimal tail comparison under convex majorization",
  "authors": [
    "Daniel J. Fresen"
  ],
  "comment": "9 pages. Most of the material here was originally part of arXiv:2203.12523 and/or arXiv:1812.10938 and now stands as a paper on its own",
  "categories": [
    "math.PR"
  ],
  "abstract": "Following results of Kemperman and Pinelis, we show that if $X$ and $Y$ are real valued random variables such that $\\mathbb{E}\\left\\vert Y\\right\\vert<\\infty$ and for all non-decreasing convex $\\varphi:\\mathbb{R}\\rightarrow [0,\\infty)$, $\\mathbb{E}\\varphi(X)\\leq\\mathbb{E}\\varphi(Y)$, then for all $s\\in\\mathbb{R}$ with $\\mathbb{P}\\left\\{Y>s\\right\\}\\neq 0$, $\\mathbb{P}\\left\\{X\\geq\\mathbb{E}\\left(Y:Y>s\\right)\\right\\}\\leq\\mathbb{P}\\left\\{Y>s\\right\\}$. This bound is sharp in essentially the strictest possible sense: for any such $Y$ and $s$ there exists such an $X$ with $\\mathbb{P}\\left\\{X\\geq \\mathbb{E}\\left(Y:Y>s\\right)\\right\\}=\\mathbb{P}\\left\\{Y>s\\right\\}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-05T08:13:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E05",
      "60E15"
    ],
    "keywords": [
      "optimal tail comparison",
      "convex majorization",
      "real valued random variables",
      "non-decreasing convex"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}