{
  "id": "math/0410097",
  "version": "v1",
  "published": "2004-10-05T16:52:39.000Z",
  "updated": "2004-10-05T16:52:39.000Z",
  "title": "Convergence of functionals of sums of r.v.s to local times of fractional stable motions",
  "authors": [
    "P. Jeganathan"
  ],
  "comment": "Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/009117904000000658",
  "journal": "Annals of Probability 2004, Vol. 32, No. 3A, 1771-1795",
  "doi": "10.1214/009117904000000658",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider a sequence X_k=\\sum_{j=0}^{\\infty}c_j\\xi_{k-j}, k\\geq 1, where c_j, j\\geq 0, is a sequence of constants and \\xi_j, -\\infty <j<\\infty, is a sequence of independent identically distributed (i.i.d.) random variables (r.v.s) belonging to the domain of attraction of a strictly stable law with index 0<\\alpha \\leq 2. Let S_k=\\sum_{j=1}^kX_j. Under suitable conditions on the constants c_j it is known that for a suitable normalizing constant \\gamma_n, the partial sum process \\gamma_n^{-1}S_{[nt]} converges in distribution to a linear fractional stable motion (indexed by \\alpha and H, 0<H<1). A fractional ARIMA process with possibly heavy tailed innovations is a special case of the process X_k. In this paper it is established that the process n^{-1}\\beta_n\\sum_{k=1}^{[nt]}f(\\beta_n(\\gamma_n^{-1}S_k+x)) converges in distribution to (\\int_{-\\infty}^{\\infty}f(y) dy)L(t,-x), where L(t,x) is the local time of the linear fractional stable motion, for a wide class of functions f(y) that includes the indicator functions of bounded intervals of the real line. Here \\beta_n\\to \\infty such that n^{-1}\\beta_n\\to 0. The only further condition that is assumed on the distribution of \\xi_1 is that either it satisfies the Cram\\'er's condition or has a nonzero absolutely continuous component. The results have motivation in large sample inference for certain nonlinear time series models.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-10-05T16:52:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "60G18",
      "60J55",
      "62M10",
      "62J02"
    ],
    "keywords": [
      "local time",
      "linear fractional stable motion",
      "convergence",
      "nonlinear time series models",
      "functionals"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.....10097J"
    }
  }
}