{
  "id": "1202.5859",
  "version": "v3",
  "published": "2012-02-27T08:51:48.000Z",
  "updated": "2013-05-23T12:45:01.000Z",
  "title": "Asympotic behavior of the total length of external branches for Beta-coalescents",
  "authors": [
    "Jean-Stephane Dhersin",
    "Linglong Yuan"
  ],
  "categories": [
    "math.PR",
    "q-bio.PE"
  ],
  "abstract": "We consider a ${\\Lambda}$-coalescent and we study the asymptotic behavior of the total length $L^{(n)}_{ext}$ of the external branches of the associated $n$-coalescent. For Kingman coalescent, i.e. ${\\Lambda}={\\delta}_0$, the result is well known and is useful, together with the total length $L^{(n)}$, for Fu and Li's test of neutrality of mutations% under the infinite sites model asumption . For a large family of measures ${\\Lambda}$, including Beta$(2-{\\alpha},{\\alpha})$ with $0<\\alpha<1$, M{\\\"o}hle has proved asymptotics of $L^{(n)}_{ext}$. Here we consider the case when the measure ${\\Lambda}$ is Beta$(2-{\\alpha},{\\alpha})$, with $1<\\alpha<2$. We prove that $n^{{\\alpha}-2}L^{(n)}_{ext}$ converges in $L^2$ to $\\alpha(\\alpha-1)\\Gamma(\\alpha)$. As a consequence, we get that $L^{(n)}_{ext}/L^{(n)}$ converges in probability to $2-\\alpha$. To prove the asymptotics of $L^{(n)}_{ext}$, we use a recursive construction of the $n$-coalescent by adding individuals one by one. Asymptotics of the distribution of $d$ normalized external branch lengths and a related moment result are also given.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-05-23T12:45:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "total length",
      "external branches",
      "asympotic behavior",
      "beta-coalescents",
      "infinite sites model asumption"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1202.5859D"
    }
  }
}