{
  "id": "1701.00549",
  "version": "v1",
  "published": "2017-01-02T22:48:36.000Z",
  "updated": "2017-01-02T22:48:36.000Z",
  "title": "The size of the last merger and time reversal in $Λ$-coalescents",
  "authors": [
    "Götz Kersting",
    "Jason Schweinsberg",
    "Anton Wakolbinger"
  ],
  "comment": "31 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the number of blocks involved in the last merger of a $\\Lambda$-coalescent started with $n$ blocks. We give conditions under which, as $n \\to \\infty$, the sequence of these random variables a) is tight, b) converges in distribution to a finite random variable or c) converges to infinity in probability. Our conditions are optimal for $\\Lambda$-coalescents that have a dust component. For general $\\Lambda$, we relate the three cases to the existence, uniqueness and non-existence of quasi-invariant measures for the dynamics of the block-counting process, and in case b) investigate the time-reversal of the block-counting process back from the time of the last merger.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-02T22:48:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J75",
      "60J27",
      "60K20"
    ],
    "keywords": [
      "time reversal",
      "coalescent",
      "block-counting process",
      "finite random",
      "random variable"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}