A note on the number of maxima in a discrete sample

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Abstract

Consider {Xj, j ≥ 1}, a sequence of i.i.d., positive, integer-valued random variables. Let Kn denote the number of the integer j ∈ {1,2,...,n} for which Xj = max1≤m≤n Xm. In this paper we prove that 1imn→∞ EKn = 1 if and only if Kn converges in probability to one, if and only if 1imn→∞ P(X1 = n)/P(X1 ≥ n)=0 and prove that Kn converges almost surely to one, if and only if ∑n=1(P(X1 = n)/P(X1 ≥ n))2 < ∞. Some of the results were shown by Baryshnikov et al. (1995) and Brands et al. (1994). AMS classification: Primary 60F15.

Original languageEnglish (US)
Pages (from-to)373-377
Number of pages5
JournalStatistics and Probability Letters
Volume33
Issue number4
StatePublished - May 16 1997

Keywords

  • Almost sure convergence
  • Maxima

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