{
  "id": "2107.06718",
  "version": "v1",
  "published": "2021-07-14T14:06:27.000Z",
  "updated": "2021-07-14T14:06:27.000Z",
  "title": "Scaling limits for the block counting process and the fixation line of a class of $Λ$-coalescents",
  "authors": [
    "Martin Möhle",
    "Benedict Vetter"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We provide scaling limits for the block counting process and the fixation line of $\\Lambda$-coalescents as the initial state $n$ tends to infinity under the assumption that the measure $\\Lambda$ on $[0,1]$ satisfies $\\int_{[0,1]}u^{-1}(\\Lambda-b\\lambda)({\\rm d}u)<\\infty$ for some $b>0$. Here $\\lambda$ denotes the Lebesgue measure. The main result states that the block counting process, properly logarithmically scaled, converges in the Skorohod space to an Ornstein--Uhlenbeck type process as $n$ tends to infinity. The result is applied to beta coalescents with parameters $1$ and $b>0$. We split the generators into two parts by additively decomposing Lambda and then prove the uniform convergence of both parts separately.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-14T14:06:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J90",
      "60J27"
    ],
    "keywords": [
      "block counting process",
      "fixation line",
      "scaling limits",
      "coalescents",
      "main result states"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}