{
  "id": "math/0204022",
  "version": "v1",
  "published": "2002-04-02T01:36:44.000Z",
  "updated": "2002-04-02T01:36:44.000Z",
  "title": "A general Hsu-Robbins-Erdos type estimate of tail probabilities of sums of independent identically distributed random variables",
  "authors": [
    "Alexander R. Pruss"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $X_1,X_2,...$ be a sequence of independent and identically distributed random variables, and put $S_n=X_1+...+X_n$. Under some conditions on the positive sequence $\\tau_n$ and the positive increasing sequence $a_n$, we give necessary and sufficient conditions for the convergence of $\\sum_{n=1}^\\infty \\tau_n P(|S_n|\\ge \\epsilon a_n)$ for all $\\epsilon>0$, generalizing Baum and Katz's (1965) generalization of the Hsu-Robbins-Erdos (1947, 1949) law of large numbers, also allowing us to characterize the convergence of the above series in the case where $\\tau_n=n^{-1}$ and $a_n=(n\\log n)^{1/2}$ for $n\\ge 2$, thereby answering a question of Spataru. Moreover, some results for non-identically distributed independent random variables are obtained by a recent comparison inequality. Our basic method is to use a central limit theorem estimate of Nagaev (1965) combined with the Hoffman-Jorgensen inequality (1974).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2002-04-02T01:36:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "60F10",
      "60F15"
    ],
    "keywords": [
      "independent identically distributed random variables",
      "general hsu-robbins-erdos type estimate",
      "tail probabilities",
      "distributed independent random variables"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math......4022P"
    }
  }
}