{
  "id": "math/0511739",
  "version": "v2",
  "published": "2005-11-30T15:31:22.000Z",
  "updated": "2007-07-25T12:14:46.000Z",
  "title": "A long range dependence stable process and an infinite variance branching system",
  "authors": [
    "Tomasz Bojdecki",
    "Luis G. Gorostiza",
    "Anna Talarczyk"
  ],
  "comment": "Published at http://dx.doi.org/10.1214/009117906000000737 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2007, Vol. 35, No. 2, 500-527",
  "doi": "10.1214/009117906000000737",
  "categories": [
    "math.PR"
  ],
  "abstract": "We prove a functional limit theorem for the rescaled occupation time fluctuations of a $(d,\\alpha,\\beta)$-branching particle system [particles moving in $\\mathbb {R}^d$ according to a symmetric $\\alpha$-stable L\\'{e}vy process, branching law in the domain of attraction of a $(1+\\beta)$-stable law, $0<\\beta<1$, uniform Poisson initial state] in the case of intermediate dimensions, $\\alpha/\\beta<d<\\alpha(1+\\beta)/\\beta$. The limit is a process of the form $K\\lambda\\xi$, where $K$ is a constant, $\\lambda$ is the Lebesgue measure on $\\mathbb {R}^d$, and $\\xi=(\\xi_t)_{t\\geq0}$ is a $(1+\\beta)$-stable process which has long range dependence. For $\\alpha<2$, there are two long range dependence regimes, one for $\\beta>d/(d+\\alpha)$, which coincides with the case of finite variance branching $(\\beta=1)$, and another one for $\\beta\\leq d/(d+\\alpha)$, where the long range dependence depends on the value of $\\beta$. The long range dependence is characterized by a dependence exponent $\\kappa$ which describes the asymptotic behavior of the codifference of increments of $\\xi$ on intervals far apart, and which is $d/\\alpha$ for the first case (and for $\\alpha=2$) and $(1+\\beta-d/(d+\\alpha))d/\\alpha$ for the second one. The convergence proofs use techniques of $\\mathcal{S}'(\\mathbb {R}^d)$-valued processes.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-07-25T12:14:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F17",
      "60J80",
      "60G18",
      "60G52"
    ],
    "keywords": [
      "long range dependence stable process",
      "infinite variance branching system",
      "long range dependence depends"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.....11739B"
    }
  }
}