{
  "id": "1601.04274",
  "version": "v1",
  "published": "2016-01-17T11:28:13.000Z",
  "updated": "2016-01-17T11:28:13.000Z",
  "title": "Functional limit theorems for the number of occupied boxes in the Bernoulli sieve",
  "authors": [
    "Gerold Alsmeyer",
    "Alexander Iksanov",
    "Alexander Marynych"
  ],
  "comment": "22 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "The Bernoulli sieve is the infinite Karlin \"balls-in-boxes\" scheme with random probabilities of stick-breaking type. Assuming that the number of placed balls equals $n$, we prove several functional limit theorems (FLTs) in the Skorohod space $D[0,1]$ endowed with the $J_{1}$- or $M_{1}$-topology for the number $K_{n}^{*}(t)$ of boxes containing at most $[n^{t}]$ balls, $t\\in[0,1]$, and the random distribution function $K_{n}^{*}(t)/K_{n}^{*}(1)$, as $n\\to\\infty$. The limit processes for $K_{n}^{*}(t)$ are of the form $(X(1)-X((1-t)-))_{t\\in[0,1]}$, where $X$ is either a Brownian motion, a spectrally negative stable L\\'evy process, or an inverse stable subordinator. The small values probabilities for the stick-breaking factor determine which of the alternatives occurs. If the logarithm of this factor is integrable, the limit process for $K_{n}^{*}(t)/K_{n}^{*}(1)$ is a L\\'evy bridge. Our approach relies upon two novel ingredients and particularly enables us to dispense with a Poissonization-de-Poissonization step which has been an essential component in all the previous studies of $K_{n}^{*}(1)$. First, for any Karlin occupancy scheme with deterministic probabilities $(p_{k})_{k\\ge 1}$, we obtain an approximation, uniformly in $t\\in[0,1]$, of the number of boxes with at most $[n^{t}]$ balls by a counting function defined in terms of $(p_{k})_{k\\ge 1}$. Second, we prove several FLTs for the number of visits to the interval $[0,nt]$ by a perturbed random walk, as $n\\to\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-17T11:28:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F17",
      "60C05"
    ],
    "keywords": [
      "functional limit theorems",
      "bernoulli sieve",
      "occupied boxes",
      "negative stable levy process",
      "limit process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160104274A"
    }
  }
}