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Limit laws for the number of near maxima via the Poisson approximation

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  • Pakes, Anthony G.
  • Li, Yun

Abstract

Given a sequence of i.i.d. random variables, new proofs are given for limit theorems for the number of observations near the maximum up to time n, as n --> [infinity]. The proofs rely on a Poisson approximation to conditioned binomial laws, and they reveal the origin in the limit laws of mixing with respect to extreme value laws. For the case of attraction to the Fréchet law, the effects of relaxing a technical condition are examined. The results are set in the broader context of counting observations near upper order statistics. This involves little extra effort.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:stapro:v:40:y:1998:i:4:p:395-401
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    References listed on IDEAS

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    1. 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.
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    Cited by:

    1. Dembinska, Anna & Iliopoulos, George, 2012. "On the asymptotics of numbers of observations in random regions determined by order statistics," Journal of Multivariate Analysis, Elsevier, vol. 103(1), pages 151-160, January.
    2. Dembinska, Anna, 2010. "On numbers of observations near randomly indexed order statistics," Statistics & Probability Letters, Elsevier, vol. 80(5-6), pages 309-317, March.
    3. Li, Y. & Pakes, Anthony G., 2001. "On the number of near-maximum insurance claims," Insurance: Mathematics and Economics, Elsevier, vol. 28(3), pages 309-323, June.
    4. Augustynowicz, Aneta, 2020. "Asymptotic behavior of proportions of observations falling to random regions determined by central order statistics," Statistics & Probability Letters, Elsevier, vol. 162(C).
    5. 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.
    6. 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.
    7. Rasbagh Vasudeva & J. Vasantha Kumari, 2014. "Asymptotic behaviour of near-maxima of Gaussian sequences," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 77(7), pages 861-866, October.
    8. 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.
    9. Hu, Zhishui & Su, Chun, 2003. "Limit theorems for the number and sum of near-maxima for medium tails," Statistics & Probability Letters, Elsevier, vol. 63(3), pages 229-237, July.
    10. M. Akbari & M. Fashandi & Jafar Ahmadi, 2016. "Characterizations based on the numbers of near-order statistics," Statistical Papers, Springer, vol. 57(1), pages 21-30, March.
    11. H. Nagaraja & Karthik Bharath & Fangyuan Zhang, 2015. "Spacings around an order statistic," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(3), pages 515-540, June.
    12. 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.
    13. 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.
    14. 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.
    15. 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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