{
  "id": "1110.2163",
  "version": "v4",
  "published": "2011-10-10T19:58:16.000Z",
  "updated": "2014-11-13T11:32:51.000Z",
  "title": "Invariance principles for Galton-Watson trees conditioned on the number of leaves",
  "authors": [
    "Igor Kortchemski"
  ],
  "comment": "46 pages, 2 figures. Published version",
  "journal": "Stochastic Process. Appl. 122 (2012), no. 9, 3126-3172",
  "doi": "10.1016/j.spa.2012.05.013",
  "categories": [
    "math.PR"
  ],
  "abstract": "We are interested in the asymptotic behavior of critical Galton-Watson trees whose offspring distribution may have infinite variance, which are conditioned on having a large fixed number of leaves. We first find an asymptotic estimate for the probability of a Galton-Watson tree having $n$ leaves. Secondly, we let $t_n$ be a critical Galton-Watson tree whose offspring distribution is in the domain of attraction of a stable law, and conditioned on having exactly $n$ leaves. We show that the rescaled Lukasiewicz path and contour function of $t_n$ converge respectively to $X^{exc}$ and $H^{exc}$, where $X^{exc}$ is the normalized excursion of a strictly stable spectrally positive L\\'evy process and $H^{exc}$ is its associated continuous-time height function. As an application, we investigate the distribution of the maximum degree in a critical Galton-Watson tree conditioned on having a large number of leaves. We also explain how these results can be generalized to the case of Galton-Watson trees which are conditioned on having a large fixed number of vertices with degree in a given set, thus extending results obtained by Aldous, Duquesne and Rizzolo.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-05-22T07:09:50.000Z",
      "title": "Invariance principles for Galton-Watson trees conditioned on their number of leaves",
      "abstract": "We are interested in the asymptotic behavior of critical Galton-Watson trees whose offspring distribution may have infinite variance, which are conditioned on having a large fixed number of leaves. We first find an asymptotic estimate for the probability of a Galton-Watson tree having $n$ leaves. Secondly, we let $\\t_n$ be a critical Galton-Watson tree whose offspring distribution is in the domain of attraction of a stable law, and conditioned on having exactly $n$ leaves. We show that the rescaled Lukasiewicz path and contour function of $\\t_n$ converge respectively to $\\X$ and $\\H$, where $\\X$ is the normalized excursion of a strictly stable spectrally positive L\\'evy process and $\\H$ is its associated continuous-time height function. As an application, we investigate the distribution of the maximum degree in a critical Galton-Watson tree conditioned on having a large number of leaves. We also explain how these results can be generalized to the case of Galton-Watson trees which are conditioned on having a large fixed number of vertices with degree in a given set, thus extending results obtained by Aldous, Duquesne and Rizzolo.",
      "comment": "46 pages, 2 figures. Final version: title changed, minor corrections. To appear in Stochastic Processes and their Applications. arXiv admin note: text overlap with arXiv:1109.4138",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2014-11-13T11:32:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "galton-watson trees",
      "invariance principles",
      "critical galton-watson tree",
      "spectrally positive levy process",
      "stable spectrally positive levy"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1110.2163K"
    }
  }
}