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A necessary and sufficient condition for the existence of the limiting probability of a tie for first place

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  • Baryshnikov, Yuliy
  • Eisenberg, Bennett
  • Stengle, Gilbert

Abstract

Suppose that the scores of n players are unbounded, independent, integer valued random variables equal in distribution to X. We show that as n --> [infinity], the limiting probability of a tie for the highest score exists if and only if P(X = j)/P(X > j) --> 0 as j --> [infinity].

Suggested Citation

  • 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.
    • Handle: RePEc:eee:stapro:v:23:y:1995:i:3:p:203-209
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    References listed on IDEAS

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    1. 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:

    1. Li, Yun, 1999. "A note on the number of records near the maximum," Statistics & Probability Letters, Elsevier, vol. 43(2), pages 153-158, June.
    2. Yakubovich, Yu., 2015. "On descents after maximal values in samples of discrete random variables," Statistics & Probability Letters, Elsevier, vol. 105(C), pages 203-208.
    3. Athreya, J. S. & Sethuraman, S., 2001. "On the asymptotics of discrete order statistics," Statistics & Probability Letters, Elsevier, vol. 54(3), pages 243-249, October.
    4. Archibald, Margaret & Knopfmacher, Arnold, 2009. "The average position of the dth maximum in a sample of geometric random variables," Statistics & Probability Letters, Elsevier, vol. 79(7), pages 864-872, April.
    5. Qi, Yongcheng & Wilms, R. J. G., 1997. "The limit behavior of maxima modulo one and the number of maxima," Statistics & Probability Letters, Elsevier, vol. 32(4), pages 357-366, April.
    6. Archibald, Margaret & Blecher, Aubrey & Brennan, Charlotte & Knopfmacher, Arnold, 2015. "Descents following maximal values in samples of geometric random variables," Statistics & Probability Letters, Elsevier, vol. 97(C), pages 229-240.
    7. 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.
    8. Olofsson, Peter, 1999. "A Poisson approximation with applications to the number of maxima in a discrete sample," Statistics & Probability Letters, Elsevier, vol. 44(1), pages 23-27, August.
    9. Eisenberg, Bennett, 2008. "On the expectation of the maximum of IID geometric random variables," Statistics & Probability Letters, Elsevier, vol. 78(2), pages 135-143, February.
    10. Arvydas Astrauskas, 2023. "Some Bounds for the Expectations of Functions on Order Statistics and Their Applications," Journal of Theoretical Probability, Springer, vol. 36(2), pages 1116-1147, June.
    11. Archibald, Margaret & Blecher, Aubrey & Brennan, Charlotte & Knopfmacher, Arnold & Prodinger, Helmut, 2017. "Geometric random variables: Descents following maxima," Statistics & Probability Letters, Elsevier, vol. 124(C), pages 140-147.
    12. Eisenberg, Bennett, 2009. "The number of players tied for the record," Statistics & Probability Letters, Elsevier, vol. 79(3), pages 283-288, February.
    13. Qi, Y. & Wilms, R. J. G., 1997. "The limit behavior of maxima modulo one and the number of maxima," Statistics & Probability Letters, Elsevier, vol. 34(1), pages 75-84, May.

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