{
  "id": "1111.1786",
  "version": "v2",
  "published": "2011-11-08T03:11:49.000Z",
  "updated": "2012-01-20T18:52:43.000Z",
  "title": "Weak Limit of the Geometric Sum of Independent But Not Identically Distributed Random Variables",
  "authors": [
    "Alexis Akira Toda"
  ],
  "categories": [
    "math.PR",
    "stat.AP"
  ],
  "abstract": "We show that when $\\set{X_j}$ is a sequence of independent (but not necessarily identically distributed) random variables which satisfies a condition similar to the Lindeberg condition, the properly normalized geometric sum $\\sum_{j=1}^{\\nu_p}X_j$ (where $\\nu_p$ is a geometric random variable with mean $1/p$) converges in distribution to a Laplace distribution as $p\\to 0$. The same conclusion holds for the multivariate case. This theorem provides a reason for the ubiquity of the double power law in economic and financial data.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-01-20T18:52:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "identically distributed random variables",
      "weak limit",
      "independent",
      "condition similar",
      "double power law"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1111.1786A"
    }
  }
}