{
  "id": "1910.13190",
  "version": "v1",
  "published": "2019-10-29T10:51:49.000Z",
  "updated": "2019-10-29T10:51:49.000Z",
  "title": "Critical branching processes in random environment and Cauchy domain of attraction",
  "authors": [
    "Congzao Dong",
    "Charline Smadi",
    "Vladimir A. Vatutin"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We are interested in the survival probability of a population modeled by a critical branching process in an i.i.d. random environment. We assume that the random walk associated with the branching process is oscillating and satisfies a Spitzer condition $\\mathbf{P}(S_{n}>0)\\rightarrow \\rho ,\\ n\\rightarrow \\infty $, which is a standard condition in fluctuation theory of random walks. Unlike the previously studied case $\\rho \\in (0,1)$, we investigate the case where the offspring distribution is in the domain of attraction of a stable law with parameter $1$, which implies that $\\rho =0$ or $1$. We find the asymptotic behaviour of the survival probability of the population in these two cases.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-29T10:51:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "critical branching process",
      "random environment",
      "cauchy domain",
      "attraction",
      "survival probability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}