{
  "id": "math/9810163",
  "version": "v1",
  "published": "1998-10-28T15:47:36.000Z",
  "updated": "1998-10-28T15:47:36.000Z",
  "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|>t a_n) for all t>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=1/n and a_n=(n log n)^{1/2} for n>1, 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": "1998-10-28T15:47:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "60F10",
      "60F15",
      "60E15"
    ],
    "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": "1998math.....10163P"
    }
  }
}