{
  "id": "2006.10401",
  "version": "v1",
  "published": "2020-06-18T10:04:29.000Z",
  "updated": "2020-06-18T10:04:29.000Z",
  "title": "Moderate parts in regenerative compositions: the case of regular variation",
  "authors": [
    "Dariusz Buraczewski",
    "Bohdan Dovgay",
    "Alexander Marynych"
  ],
  "comment": "15 pages, 1 figure",
  "categories": [
    "math.PR"
  ],
  "abstract": "A regenerative random composition of integer $n$ is constructed by allocating $n$ standard exponential points over a countable number of intervals, comprising the complement of the closed range of a subordinator $S$. Assuming that the L\\'{e}vy measure of $S$ is infinite and regularly varying at zero of index $-\\alpha$, $\\alpha\\in(0,\\,1)$, we find an explicit threshold $r=r(n)$, such that the number $K_{n,\\,r(n)}$ of blocks of size $r(n)$ converges in distribution without any normalization to a mixed Poisson distribution. The sequence $(r(n))$ turns out to be regularly varying with index $\\alpha/(\\alpha+1)$ and the mixing distribution is that of the exponential functional of $S$. We also discuss asymptotic behavior of $K_{n,\\,w(n)}$ in cases when $w(n)$ diverges but grows slower than $r(n)$. Our findings complement previously known strong laws of large numbers for $K_{n,\\,r}$ in case of a fixed $r\\in\\N$. As a key tool we employ new Abelian theorems for Laplace--Stiletjes transforms of regularly varying functions with the indexes of regular variation diverging to infinity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-18T10:04:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60C05",
      "60F05"
    ],
    "keywords": [
      "regular variation",
      "moderate parts",
      "regenerative compositions",
      "standard exponential points",
      "regularly varying"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}