A clustering law for some discrete order statistics

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3 Scopus citations


Let X1, X2, ..., Xn be a sequence of independent, identically distributed positive integer random variables with distribution function F. Anderson (1970) proved a variant of the law of large numbers by showing that the sample maximum moves asymptotically on two values if and only if F satisfies a 'clustering' condition, limn → ∞[1 -F(n + 1)]/[1 - F(n)] = 0. In this article, we generalize Anderson's result and show that it is robust by proving that, for any r ≥ 0, the sample maximum and other extremes asymptotically cluster on r + 2 values if and only if limn → ∞ [1 - F(n + r + 1)]/[1 - F(n)] = 0. Together with previous work which considered other asymptotic properties of these sample extremes, a more detailed asymptotic clustering structure for discrete order statistics is presented.

Original languageEnglish (US)
Pages (from-to)226-241
Number of pages16
JournalJournal of Applied Probability
Issue number1
StatePublished - Mar 2003
Externally publishedYes


  • Asymptotics
  • Clustering
  • Extremes
  • Law of large numbers
  • Order statistics

ASJC Scopus subject areas

  • Statistics and Probability
  • Mathematics(all)
  • Statistics, Probability and Uncertainty


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