IDEAS home Printed from https://ideas.repec.org/a/eee/stapro/v35y1997i1p49-58.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167-7152(96)00215-5
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Jasiński, Krzysztof, 2016. "Asymptotic normality of numbers of observations near order statistics from stationary processes," Statistics & Probability Letters, Elsevier, vol. 119(C), pages 259-263.
    2. 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.
    3. Eisenberg, Bennett, 2009. "The number of players tied for the record," Statistics & Probability Letters, Elsevier, vol. 79(3), pages 283-288, February.
    4. Li, Yun, 1999. "A note on the number of records near the maximum," Statistics & Probability Letters, Elsevier, vol. 43(2), pages 153-158, June.
    5. 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.
    6. 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.
    7. Nevzorov, Valery B. & Stepanov, Alexei, 2024. "On uppermost discrete spacing," Statistics & Probability Letters, Elsevier, vol. 208(C).
    8. 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.
    9. 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.
    10. 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.
    11. 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.
    12. Lin, Jin-Guan & Huang, Chao & Zhuang, Qing-Yun & Zhu, Li-Ping, 2010. "Estimating generalized state density of near-extreme events and its applications in analyzing stock data," Insurance: Mathematics and Economics, Elsevier, vol. 47(1), pages 13-20, August.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:stapro:v:35:y:1997:i:1:p:49-58. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.