{
  "id": "1305.6522",
  "version": "v1",
  "published": "2013-05-28T15:03:22.000Z",
  "updated": "2013-05-28T15:03:22.000Z",
  "title": "Probability Distribution Function for the Euclidean Distance Between Two Telegraph Processes",
  "authors": [
    "Alexander D. Kolesnik"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider two independent Goldstein-Kac telegraph processes $X_1(t)$ and $X_2(t)$ on the real line $\\Bbb R$. The processes $X_k(t), \\; k=1,2,$ are performed by stochastic motions at finite constant velocities $c_1>0, \\; c_2>0,$ that start at the initial time instant $t=0$ from the origin of the real line $\\Bbb R$ and are controlled by two independent homogeneous Poisson processes of rates $\\lambda_1>0, \\; \\lambda_2>0$, respectively. Closed-form expression for the probability distribution function of the Euclidean distance $$\\rho(t) = |X_1(t) - X_2(t) |, \\qquad t>0,$$ between these processes at arbitrary time instant $t>0$, is obtained. Some numerical results are presented.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-05-28T15:03:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "60J60",
      "60J65",
      "82C41",
      "82C70"
    ],
    "keywords": [
      "probability distribution function",
      "euclidean distance",
      "independent goldstein-kac telegraph processes",
      "real line",
      "finite constant velocities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1305.6522K"
    }
  }
}