{
  "id": "2109.13315",
  "version": "v2",
  "published": "2021-09-27T19:15:21.000Z",
  "updated": "2022-08-16T07:26:08.000Z",
  "title": "Critical branching processes in random environment with immigration: the size of the only surviving family",
  "authors": [
    "Charline Smadi",
    "Vladimir A. Vatutin"
  ],
  "comment": "version, as published in the Proc. Steklov Inst. Math. arXiv admin note: text overlap with arXiv:1911.00316",
  "journal": "Proc. Steklov Inst. Math. 316 (2022)",
  "doi": "10.1134/S0081543822010230",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider a critical branching process $Y_{n}$ in an i.i.d. random environment, in which one immigrant arrives at each generation. Let $% \\mathcal{A}_{i}(n)$ be the event that all individuals alive at time $n$ are offspring of the immigrant which joined the population at time $i$. We study the conditional distribution of $Y_{n}$ given $\\mathcal{A}_{i}(n)$ 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 $n$).",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-08-16T07:26:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "critical branching process",
      "random environment",
      "surviving family",
      "immigration",
      "immigrant arrives"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}