{
  "id": "0904.4127",
  "version": "v2",
  "published": "2009-04-27T11:29:46.000Z",
  "updated": "2009-11-24T14:20:32.000Z",
  "title": "Limiting Distributions for Sums of Independent Random Products",
  "authors": [
    "Zakhar Kabluchko"
  ],
  "comment": "31 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\{X_{i,j}:(i,j)\\in\\mathbb N^2\\}$ be a two-dimensional array of independent copies of a random variable $X$, and let $\\{N_n\\}_{n\\in\\mathbb N}$ be a sequence of natural numbers such that $\\lim_{n\\to\\infty}e^{-cn}N_n=1$ for some $c>0$. Our main object of interest is the sum of independent random products $$Z_n=\\sum_{i=1}^{N_n} \\prod_{j=1}^{n}e^{X_{i,j}}.$$ It is shown that the limiting properties of $Z_n$, as $n\\to\\infty$, undergo phase transitions at two critical points $c=c_1$ and $c=c_2$. Namely, if $c>c_2$, then $Z_n$ satisfies the central limit theorem with the usual normalization, whereas for $c<c_2$, a totally skewed $\\alpha$-stable law appears in the limit. Further, $Z_n/\\mathbb E Z_n$ converges in probability to 1 if and only if $c>c_1$. If the random variable $X$ is Gaussian, we recover the results of Bovier, Kurkova, and L\\\"owe [Fluctuations of the free energy in the REM and the $p$-spin SK models. Ann. Probab. 30(2002), 605-651].",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-11-24T14:20:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G50",
      "60F05",
      "60F10"
    ],
    "keywords": [
      "independent random products",
      "limiting distributions",
      "spin sk models",
      "undergo phase transitions",
      "central limit theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0904.4127K"
    }
  }
}