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Asymptotic behaviour of a number of repeated records

Author

Listed:
  • Khmaladze, E.
  • Nadareishvili, M.
  • Nikabadze, A.

Abstract

Given a sequence of i.i.d. random variables with continuous distribution function we study the number of [var epsilon]-repetitions of the current record value from the time it occurs up to some n, as n --> [infinity]. This number of repetitions typically does not converge, but oscillates indefinitely. However, if it does converge, it can do so only to 1 or to [infinity]. For deterministic choices of [var epsilon], we give exact and asymptotic distributions and set conditions on the tail of F for convergence in probability to 1 and to [infinity] and for a.s. convergence to 1. For one choice of [var epsilon] as a radom variable, we obtain a never converging sequence of [var epsilon]-repetitions with limiting distribution being the so called Zipf's law, famous in an entirely different context.

Suggested Citation

  • Khmaladze, E. & Nadareishvili, M. & Nikabadze, A., 1997. "Asymptotic behaviour of a number of repeated records," Statistics & Probability Letters, Elsevier, vol. 35(1), pages 49-58, August.
    • Handle: RePEc:eee:stapro:v:35:y:1997:i:1:p:49-58
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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. Yun Li & Quanxi Shao, 2007. "Slow convergence of the number of near-maxima for Burr XII distributions," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 66(1), pages 89-104, July.
    2. A. Stepanov, 2007. "The number of records within a random interval of the current record value," Statistical Papers, Springer, vol. 48(1), pages 63-79, January.
    3. Hashorva, Enkelejd & Hüsler, Jürg, 2001. "On the number of points near the multivariate maxima," Statistics & Probability Letters, Elsevier, vol. 55(2), pages 113-124, November.
    4. Raúl Gouet & F. López & Gerardo Sanz, 2012. "On δ-record observations: asymptotic rates for the counting process and elements of maximum likelihood estimation," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(1), pages 188-214, March.
    5. Pakes, Anthony G. & Li, Yun, 1998. "Limit laws for the number of near maxima via the Poisson approximation," Statistics & Probability Letters, Elsevier, vol. 40(4), pages 395-401, November.
    6. A. Castaño-Martínez & F. López-Blázquez & B. Salamanca-Miño, 2016. "Exceedances of records," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 79(7), pages 837-866, October.
    7. 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.
    8. Bairamov, I. & Stepanov, A., 2010. "Numbers of near-maxima for the bivariate case," Statistics & Probability Letters, Elsevier, vol. 80(3-4), pages 196-205, February.
    9. Hashorva, Enkelejd, 2003. "On the number of near-maximum insurance claim under dependence," Insurance: Mathematics and Economics, Elsevier, vol. 32(1), pages 37-49, February.

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