{
  "id": "2106.11249",
  "version": "v1",
  "published": "2021-06-21T16:47:10.000Z",
  "updated": "2021-06-21T16:47:10.000Z",
  "title": "Branching in a Markovian Environment",
  "authors": [
    "Lila Greco",
    "Lionel Levine"
  ],
  "comment": "26 pages, 1 figure",
  "categories": [
    "math.PR"
  ],
  "abstract": "A branching process in a Markovian environment consists of an irreducible Markov chain on a set of \"environments\" together with an offspring distribution for each environment. At each time step the chain transitions to a new random environment, and one individual is replaced by a random number of offspring whose distribution depends on the new environment. We give a first moment condition that determines whether this process survives forever with positive probability. On the event of survival we prove a law of large numbers and a central limit theorem for the population size. We also define a matrix-valued generating function for which the extinction matrix (whose entries are the probability of extinction in state j given that the initial state is i) is a fixed point, and we prove that iterates of the generating function starting with the zero matrix converge to the extinction matrix.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-21T16:47:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J80",
      "60J10",
      "60K37",
      "60F05",
      "15A24",
      "15B51"
    ],
    "keywords": [
      "extinction matrix",
      "process survives forever",
      "markovian environment consists",
      "central limit theorem",
      "generating function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}