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A note on the number of maxima in a discrete sample

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  • Qi, Yongcheng

Abstract

Consider {Xj, j [greater-or-equal, slanted] 1}, a sequence of i.i.d., positive, integer-valued random variables. Let Kn denote the number of the integer j [set membership, variant] {1,2,...,n} for which Xj = max1[less-than-or-equals, slant]m[less-than-or-equals, slant]n Xm. In this paper we prove that limn-->[infinity] EKn = 1 if and only if Kn converges in probability to one, if and only if limn-->[infinity] P(X1=n)/P(X1[greater-or-equal, slanted]n)=0 and prove that Kn converges almost surely to one, if and only if . Some of the results were shown by Baryshnikov et al. (1995) and Brands et al. (1994).

Suggested Citation

  • Qi, Yongcheng, 1997. "A note on the number of maxima in a discrete sample," Statistics & Probability Letters, Elsevier, vol. 33(4), pages 373-377, May.
    • Handle: RePEc:eee:stapro:v:33:y:1997:i:4:p:373-377
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    References listed on IDEAS

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    1. Baryshnikov, Yuliy & Eisenberg, Bennett & Stengle, Gilbert, 1995. "A necessary and sufficient condition for the existence of the limiting probability of a tie for first place," Statistics & Probability Letters, Elsevier, vol. 23(3), pages 203-209, May.
    2. Brands, J. J. A. M. & Steutel, F. W. & Wilms, R. J. G., 1994. "On the number of maxima in a discrete sample," Statistics & Probability Letters, Elsevier, vol. 20(3), pages 209-217, June.
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    Cited by:

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    3. Balakrishnan, N. & Stepanov, A., 2004. "A note on the paper of Khmaladze et al," Statistics & Probability Letters, Elsevier, vol. 68(4), pages 415-419, July.
    4. Anna Dembińska, 2017. "An ergodic theorem for proportions of observations that fall into random sets determined by sample quantiles," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 80(3), pages 319-332, April.

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    Keywords

    Maxima Almost sure convergence;

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