{
  "id": "1309.5928",
  "version": "v3",
  "published": "2013-09-23T19:29:25.000Z",
  "updated": "2015-02-18T17:32:16.000Z",
  "title": "Characteristic function of the positive part of a random variable and related results, with applications",
  "authors": [
    "Iosif Pinelis"
  ],
  "comment": "2 pages. Version 2: 3 pages; a formula for the c.f. of |X| and three references are added; discussion expanded; title and abstract changed; Version 3: 9 pages; the paper is completely reworked; title and abstract changed; applications are given; in particular, a more explicit and compact form of Spitzer's identity is presented",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $X$ be an arbitrary real-valued random variable (r.v.), with the characteristic function (c.f.) $f$. Integral expressions for the c.f.\\ of the r.v.'s $\\max(0,X)$ in terms of $f$ are given, as well as other related results. Applications to stock options and random walks are presented. In particular, a more explicit and compact form of Spitzer's identity is obtained.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-09-29T18:59:58.000Z",
      "title": "On the characteristic functions of the positive part and absolute value of a random variable",
      "abstract": "Let X be an arbitrary real-valued random variable (r.v.), with the characteristic function (c.f.) f. Expressions for the c.f.'s of the r.v.'s \\max(0,X) and |X| in terms of (f and) the Hilbert transform of f are given.",
      "comment": "2 pages. Version 2: 3 pages; a formula for the c.f. of |X| and three references are added; discussion expanded; title and abstract changed",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-02-18T17:32:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E10",
      "44A15",
      "60G50",
      "90B22",
      "91G20"
    ],
    "keywords": [
      "characteristic function",
      "absolute value",
      "positive part",
      "hilbert transform",
      "arbitrary real-valued random variable"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 2,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1309.5928P"
    }
  }
}