{
  "id": "1212.5434",
  "version": "v1",
  "published": "2012-12-21T13:51:21.000Z",
  "updated": "2012-12-21T13:51:21.000Z",
  "title": "Fluctuations for the number of records on subtrees of the Continuum Random Tree",
  "authors": [
    "Patrick Hoscheit"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the asymptotic behavior af the number of cuts $X(T_n)$ needed to isolate the root in a rooted binary random tree $T_n$ with $n$ leaves. We focus on the case of subtrees of the Continuum Random Tree generated by uniform sampling of leaves. We elaborate on a recent result by Abraham and Delmas, who showed that $X(T_n)/\\sqrt{2n}$ converges a.s. towards a Rayleigh-distributed random variable $\\Theta$, which gives a continuous analog to an earlier result by Janson on conditioned, finite-variance Galton-Watson trees. We prove a convergence in distribution of $n^{-1/4}(X(T_n)-\\sqrt{2n}\\Theta)$ towards a random mixture of Gaussian variables. The proofs use martingale limit theory for random processes defined on the CRT, related to the theory of records of Poisson point processes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-12-21T13:51:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "continuum random tree",
      "fluctuations",
      "poisson point processes",
      "asymptotic behavior af",
      "martingale limit theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1212.5434H"
    }
  }
}