{
  "id": "1402.6866",
  "version": "v1",
  "published": "2014-02-27T11:33:02.000Z",
  "updated": "2014-02-27T11:33:02.000Z",
  "title": "The explicit probability distribution of the sum of two telegraph processes",
  "authors": [
    "Alexander D. Kolesnik"
  ],
  "comment": "28 pages, 2 figures. arXiv admin note: text overlap with arXiv:1305.6522",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider two independent Goldstein-Kac telegraph processes $X_1(t)$ and $X_2(t)$ on the real line $\\Bbb R$, both developing with finite constant speed $c>0$, that, at the initial time instant $t=0$, simultaneously start from the origin $0\\in\\Bbb R$ and whose evolutions are controlled by two independent homogeneous Poisson processes of the same rate $\\lambda>0$. Closed-form expressions for the transition density $p(x,t)$ and the probability distribution function $\\Phi(x,t)=\\text{Pr} \\{ S(t)<x \\}, \\; x\\in\\Bbb R, \\; t>0,$ of the sum $S(t)=X_1(t)+X_2(t)$ of these processes at arbitrary time instant $t>0$, are obtained. It is also proved that the shifted time derivative $g(x,t)=(\\partial/\\partial t+2\\lambda)p(x,t)$ satisfies the Goldstein-Kac telegraph equation with doubled parameters $2c$ and $2\\lambda$. From this fact it follows that $p(x,t)$ solves a third-order hyperbolic partial differential equation, but is not its fundamental solution. The general case is also discussed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-02-27T11:33:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "60K99",
      "60J60",
      "60J65",
      "82C41",
      "82C70"
    ],
    "keywords": [
      "explicit probability distribution",
      "third-order hyperbolic partial differential equation",
      "independent goldstein-kac telegraph processes",
      "goldstein-kac telegraph equation",
      "arbitrary time instant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.6866K"
    }
  }
}