{
  "id": "1204.5039",
  "version": "v1",
  "published": "2012-04-23T12:16:33.000Z",
  "updated": "2012-04-23T12:16:33.000Z",
  "title": "Record Statistics for Multiple Random Walks",
  "authors": [
    "Gregor Wergen",
    "Satya N. Majumdar",
    "Gregory Schehr"
  ],
  "comment": "25 pages, 8 figures",
  "journal": "Phys. Rev. E 86, 011119 (2012)",
  "doi": "10.1103/PhysRevE.86.011119",
  "categories": [
    "cond-mat.stat-mech",
    "cond-mat.dis-nn",
    "math.PR",
    "physics.data-an",
    "q-fin.ST"
  ],
  "abstract": "We study the statistics of the number of records R_{n,N} for N identical and independent symmetric discrete-time random walks of n steps in one dimension, all starting at the origin at step 0. At each time step, each walker jumps by a random length drawn independently from a symmetric and continuous distribution. We consider two cases: (I) when the variance \\sigma^2 of the jump distribution is finite and (II) when \\sigma^2 is divergent as in the case of L\\'evy flights with index 0 < \\mu < 2. In both cases we find that the mean record number <R_{n,N}> grows universally as \\sim \\alpha_N \\sqrt{n} for large n, but with a very different behavior of the amplitude \\alpha_N for N > 1 in the two cases. We find that for large N, \\alpha_N \\approx 2 \\sqrt{\\log N} independently of \\sigma^2 in case I. In contrast, in case II, the amplitude approaches to an N-independent constant for large N, \\alpha_N \\approx 4/\\sqrt{\\pi}, independently of 0<\\mu<2. For finite \\sigma^2 we argue, and this is confirmed by our numerical simulations, that the full distribution of (R_{n,N}/\\sqrt{n} - 2 \\sqrt{\\log N}) \\sqrt{\\log N} converges to a Gumbel law as n \\to \\infty and N \\to \\infty. In case II, our numerical simulations indicate that the distribution of R_{n,N}/\\sqrt{n} converges, for n \\to \\infty and N \\to \\infty, to a universal nontrivial distribution, independently of \\mu. We discuss the applications of our results to the study of the record statistics of 366 daily stock prices from the Standard & Poors 500 index.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-04-23T12:16:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05.40.-a",
      "02.50.Sk"
    ],
    "keywords": [
      "multiple random walks",
      "record statistics",
      "independent symmetric discrete-time random walks",
      "random length drawn",
      "numerical simulations"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "APS",
      "journal": "Physical Review E",
      "year": 2012,
      "month": "Jul",
      "volume": 86,
      "number": 1,
      "pages": "011119"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012PhRvE..86a1119W"
    }
  }
}