{
  "id": "1003.1657",
  "version": "v1",
  "published": "2010-03-08T15:49:07.000Z",
  "updated": "2010-03-08T15:49:07.000Z",
  "title": "Limit laws for sums of independent random products: the lattice case",
  "authors": [
    "Zakhar Kabluchko"
  ],
  "comment": "12 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\{V_{i,j}; (i,j)\\in\\N^2\\}$ be a two-dimensional array of i.i.d.\\ random variables. The limit laws of the sum of independent random products $$ Z_n=\\sum_{i=1}^{N_n} \\prod_{j=1}^{n} e^{V_{i,j}} $$ as $n,N_n\\to\\infty$ have been investigated by a number of authors. Depending on the growth rate of $N_n$, the random variable $Z_n$ obeys a central limit theorem, or has limiting $\\alpha$-stable distribution. The latter result is true for non-lattice $V_{i,j}$ only. Our aim is to study the lattice case. We prove that although the (suitably normalized) sequence $Z_n$ fails to converge in distribution, it is relatively compact in the weak topology, and describe its cluster set. This set is a topological circle consisting of semi-stable distributions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-03-08T15:49:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G50",
      "60F05",
      "60F10"
    ],
    "keywords": [
      "independent random products",
      "limit laws",
      "lattice case",
      "distribution",
      "central limit theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1003.1657K"
    }
  }
}