{
  "id": "math/0511745",
  "version": "v1",
  "published": "2005-11-30T16:19:50.000Z",
  "updated": "2005-11-30T16:19:50.000Z",
  "title": "Occupation time fluctuations of an infinite variance branching system in large dimensions",
  "authors": [
    "Tomasz Bojdecki",
    "Luis G. Gorostiza",
    "Anna Talarczyk"
  ],
  "comment": "18 pages",
  "journal": "Bernoulli 13 (2007), no. 1, 20-39",
  "doi": "10.3150/07-BEJ5170",
  "categories": [
    "math.PR"
  ],
  "abstract": "We prove limit theorems for rescaled occupation time fluctuations of a (d,alpha,beta)-branching particle system (particles moving in R^d according to a spherically symmetric alpha-stable Levy process, (1+beta)-branching, 0<beta<1, uniform Poisson initial state), in the cases of critical dimension, d=alpha(1+beta)/beta, and large dimensions, d>alpha(1+beta)/beta. The fluctuation processes are continuous but their limits are stable processes with independent increments, which have jumps. The convergence is in the sense of finite-dimensional distributions, and also of space-time random fields (tightness does not hold in the usual Skorohod topology). The results are in sharp contrast with those for intermediate dimensions, alpha/beta < d < d(1+beta)/beta, where the limit process is continuous and has long range dependence (this case is studied by Bojdecki et al, 2005). The limit process is measure-valued for the critical dimension, and S'(R^d)-valued for large dimensions. We also raise some questions of interpretation of the different types of dimension-dependent results obtained in the present and previous papers in terms of properties of the particle system.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-11-30T16:19:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F17",
      "60J80",
      "60G18",
      "60G52"
    ],
    "keywords": [
      "occupation time fluctuations",
      "infinite variance branching system",
      "large dimensions",
      "symmetric alpha-stable levy process",
      "particle system"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.....11745B"
    }
  }
}