{
  "id": "1107.2855",
  "version": "v3",
  "published": "2011-07-14T15:40:42.000Z",
  "updated": "2012-10-19T12:41:01.000Z",
  "title": "The asymptotic distribution of the length of Beta-coalescent trees",
  "authors": [
    "Götz Kersting"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/11-AAP827 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Applied Probability 2012, Vol. 22, No. 5, 2086-2107",
  "doi": "10.1214/11-AAP827",
  "categories": [
    "math.PR"
  ],
  "abstract": "We derive the asymptotic distribution of the total length $L_n$ of a $\\operatorname {Beta}(2-\\alpha,\\alpha)$-coalescent tree for $1<\\alpha<2$, starting from $n$ individuals. There are two regimes: If $\\alpha\\le1/2(1+\\sqrt{5})$, then $L_n$ suitably rescaled has a stable limit distribution of index $\\alpha$. Otherwise $L_n$ just has to be shifted by a constant (depending on $n$) to get convergence to a nondegenerate limit distribution. As a consequence, we obtain the limit distribution of the number $S_n$ of segregation sites. These are points (mutations), which are placed on the tree's branches according to a Poisson point process with constant rate.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-10-19T12:41:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "asymptotic distribution",
      "beta-coalescent trees",
      "poisson point process",
      "nondegenerate limit distribution",
      "constant rate"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1107.2855K"
    }
  }
}