{
  "id": "1801.04606",
  "version": "v1",
  "published": "2018-01-14T19:52:08.000Z",
  "updated": "2018-01-14T19:52:08.000Z",
  "title": "A functional limit theorem for the profile of random recursive trees",
  "authors": [
    "Alexander Iksanov",
    "Zakhar Kabluchko"
  ],
  "comment": "10 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $X_n(k)$ be the number of vertices at level $k$ in a random recursive tree with $n+1$ vertices. We prove a functional limit theorem for the vector-valued process $(X_{[n^t]}(1),\\ldots, X_{[n^t]}(k))_{t\\geq 0}$, for each $k\\in\\mathbb N$. We show that after proper centering and normalization, this process converges weakly to a vector-valued Gaussian process whose components are integrated Brownian motions. This result is deduced from a functional limit theorem for Crump-Mode-Jagers branching processes generated by increasing random walks with increments that have finite second moment.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-14T19:52:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F17",
      "60J80",
      "60G50",
      "60C05",
      "60F05"
    ],
    "keywords": [
      "functional limit theorem",
      "random recursive tree",
      "finite second moment",
      "increasing random walks",
      "process converges"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}