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Distributions of the Number of Records and the Waiting Time Distributions for the rth Record

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  • Yung-Ming Chang

    (National Taitung University)

  • James C. Fu

    (University of Manitoba)

  • Tung-Lung Wu

    (Mississippi State University)

Abstract

Consider a sequence of independent and identically distributed random variables { $$X_t$$ X t } $$_{t = 1}^{\infty }$$ t = 1 ∞ defined on a finite state space. Our goal in this paper is to investigate the exact distributions of records associated with { $$X_t$$ X t } $$_{t = 1}^{\infty }$$ t = 1 ∞ . Based on the finite Markov chain imbedding technique, we propose a comprehensive approach to obtain the exact joint distribution of the number of records and their record values up to time n, the distribution of waiting time for the rth record, and the conditional distributions of waiting times of inter-records. In addition, we extend the results to the case where the underlying sequence { $$X_t$$ X t } $$_{t = 1}^{\infty }$$ t = 1 ∞ has an infinite state space. Examples are provided to illustrate the proposed method.

Suggested Citation

  • Yung-Ming Chang & James C. Fu & Tung-Lung Wu, 2025. "Distributions of the Number of Records and the Waiting Time Distributions for the rth Record," Methodology and Computing in Applied Probability, Springer, vol. 27(3), pages 1-13, September.
    • Handle: RePEc:spr:metcap:v:27:y:2025:i:3:d:10.1007_s11009-025-10190-6
    DOI: 10.1007/s11009-025-10190-6
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    References listed on IDEAS

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    1. Vervaat, Wim, 1973. "Limit theorems for records from discrete distributions," Stochastic Processes and their Applications, Elsevier, vol. 1(4), pages 317-334, October.
    2. Bunge, J. A. & Nagaraja, H. N., 1991. "The distributions of certain record statistics from a random number of observations," Stochastic Processes and their Applications, Elsevier, vol. 38(1), pages 167-183, June.
    3. Resnick, Sidney I., 1973. "Limit laws for record values," Stochastic Processes and their Applications, Elsevier, vol. 1(1), pages 67-82, January.
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