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Limit theorems for records from discrete distributions

Author

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  • Vervaat, Wim

Abstract

Weak and strong functional limit theorems are obtained for record values and record epochs in a sequence of independent random variables with common distribution F. The emphasis is on the case in which F is concentrated on the non-negative integers. For contrast, the well-known case of continuous F is also considered. Analogues of results obtained earlier by Resnick, de Haan and the author for continuous F are presented here for F concentrated on the non-negative integers. Also is investigated under which circumstances the latter case is so close to the continuous F case that the resulting limit theorems are the same.

Suggested Citation

  • Vervaat, Wim, 1973. "Limit theorems for records from discrete distributions," Stochastic Processes and their Applications, Elsevier, vol. 1(4), pages 317-334, October.
    • Handle: RePEc:eee:spapps:v:1:y:1973:i:4:p:317-334
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    Citations

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    Cited by:

    1. Eric S. Key, 2005. "On the Number of Records in an iid Discrete Sequence," Journal of Theoretical Probability, Springer, vol. 18(1), pages 99-107, April.
    2. Miguel Lafuente & Raúl Gouet & F. Javier López & Gerardo Sanz, 2022. "Near-Record Values in Discrete Random Sequences," Mathematics, MDPI, vol. 10(14), pages 1-20, July.
    3. Dembinska, A. & Stepanov, A., 2006. "Limit theorems for the ratio of weak records," Statistics & Probability Letters, Elsevier, vol. 76(14), pages 1454-1464, August.
    4. Krzysztof Jasiński, 2018. "Relations for product moments and covariances of kth records from discrete distributions," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 81(2), pages 125-141, February.
    5. 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.
    6. Fernando López-Blázquez & Jack Wesołowski, 2001. "Discrete distributions for which the regression of the first record on the second is linear," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 10(1), pages 121-131, June.
    7. Manuel Franco & José Ruiz, 2001. "Characterizations of discrete distributions based on conditional expectations of record values," Statistical Papers, Springer, vol. 42(1), pages 101-110, January.
    8. Gouet, Raúl & Javier López, F. & Sanz, Gerardo, 2008. "Laws of large numbers for the number of weak records," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2010-2017, October.
    9. Clément Dombry & Michael Falk & Maximilian Zott, 2019. "On Functional Records and Champions," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1252-1277, September.
    10. Enkelejd Hashorva & Alexei Stepanov, 2012. "Limit theorems for the spacings of weak records," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 75(2), pages 163-180, February.
    11. Dembinska, Anna & López-Blázquez, Fernando, 2005. "kth records from discrete distributions," Statistics & Probability Letters, Elsevier, vol. 71(3), pages 203-214, March.
    12. Pakhteev, A. & Stepanov, A., 2019. "Discrete records: Limit theorems for their spacings and generation methods," Statistics & Probability Letters, Elsevier, vol. 148(C), pages 134-142.
    13. Bairamov, Ismihan & Stepanov, Alexei, 2006. "A note on large deviations for weak records," Statistics & Probability Letters, Elsevier, vol. 76(14), pages 1449-1453, August.
    14. López-Blázquez, F. & Salamanca Miño, B. & Dembinska, A., 2005. "A note on the distribution of kth records from discrete distributions," Statistics & Probability Letters, Elsevier, vol. 75(4), pages 325-330, December.

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