{
  "id": "1911.00316",
  "version": "v1",
  "published": "2019-11-01T12:10:38.000Z",
  "updated": "2019-11-01T12:10:38.000Z",
  "title": "Critical branching processes in random environment with immigration: survival of a single family",
  "authors": [
    "Charline Smadi",
    "Vladimir A. Vatutin"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider a critical branching process in an i.i.d. random environment, in which one immigrant arrives at each generation. We are interested in the event $\\mathcal{A}_i(n)$ that all individuals alive at time $n$ are offspring of the immigrant which joined the population at time $i$. We study the asymptotic probability of this event when $n$ is large and $i$ follows different asymptotics which may be related to $n$ ($i$ fixed, close to $n$, or going to infinity but far from $n$). In order to do so, we establish some conditional limit theorems for random walks, which are of independent interest.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-01T12:10:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "critical branching process",
      "random environment",
      "single family",
      "immigration",
      "conditional limit theorems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}