{
  "id": "2308.01739",
  "version": "v1",
  "published": "2023-08-03T13:00:52.000Z",
  "updated": "2023-08-03T13:00:52.000Z",
  "title": "Records in the Infinite Occupancy Scheme",
  "authors": [
    "Zakaria Derbazi",
    "Alexander Gnedin",
    "Alexander Marynych"
  ],
  "comment": "23 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the classic infinite occupancy scheme, where balls are thrown in boxes independently, with probability $p_j$ of hitting box $j$. Each time a box receives its first ball we speak of a record and, more generally, call an $r$-record every event when a box receives its $r$th ball. Assuming that the sequence $(p_j)$ is not decaying too fast, we show that after many balls have been thrown, the suitably scaled point process of $r$-record times is approximately Poisson. The joint convergence of $r$-record processes is argued under a condition of regular variation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-03T13:00:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60C05",
      "60F05",
      "60G55"
    ],
    "keywords": [
      "box receives",
      "classic infinite occupancy scheme",
      "regular variation",
      "th ball",
      "suitably scaled point process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}