{
  "id": "2207.10923",
  "version": "v1",
  "published": "2022-07-22T08:00:15.000Z",
  "updated": "2022-07-22T08:00:15.000Z",
  "title": "The coalescent structure of Galton-Watson trees in varying environments",
  "authors": [
    "Harris Simon C.",
    "Palau Sandra",
    "Pardo Juan Carlos"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We investigate the genealogy of a sample of k particles chosen uniformly without replacement from a population alive at a large time n in a critical discrete-time Galton-Watson process in a varying environment (GWVE). We will show that subject to an explicit deterministic time-change involving only the mean and variances of the varying offspring distributions, the sample genealogy always converges to the same universal genealogical structure; it has the same tree topology as Kingman's coalescent, and the coalescent times of the k-1 pairwise mergers look like a mixture of independent identically distributed times. Our approach uses k spine particles and a change of measure $\\mathbf{Q}_n^{(e,k, \\theta)}$ under which the spines form a uniform sample without replacement at time n, as required, but additionally there is k-size biasing and discounting at rate $\\theta$ by the population size at time n. We give a forward in time construction for the process with k spines under $\\mathbf{Q}_n^{(e,k, \\theta)}$ where the k size-biasing provides a great deal of structural independence that enables explicit calculations, and the discounting means that no additional offspring moment assumptions are required. Combining the special properties of $\\mathbf{Q}_n^{(e,k, \\theta)}$ together with the Yaglom theorem for GWVE, plus a suitable time-change that encodes variation within the environment, we then undo the k-size biasing and discounting to recover the limiting genealogy for a uniform sample of size k in the GWVE. Our work extends the spine techniques developed in Harris, et. al [Annals of Applied Probability, 2020] and complements recent work by Kersting [Proceedings of the Steklov Institute of Mathematics, 2022] which describes the genealogy of the entire extant population in the GWVE in the large time limit",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-22T08:00:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J80",
      "60F17",
      "60G09"
    ],
    "keywords": [
      "varying environment",
      "galton-watson trees",
      "coalescent structure",
      "uniform sample",
      "critical discrete-time galton-watson process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}