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On the Longest Run and the Waiting Time for the First Run in a Continuous Time Multi-State Markov Chain

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  • Eutichia Vaggelatou

    (National and Kapodistrian University of Athens)

Abstract

In this paper, a marked point process with $$r+1$$ r + 1 types of marks (r types of successes $$S_{1},S_{2},\ldots ,S_{r}$$ S 1 , S 2 , … , S r and a failure F), $$r\ge 1$$ r ≥ 1 , that appear in continuous time according to a continuous-time Markov chain is considered. By constructing an appropriate embedded process using Markov chain embedding technique in continuous time, the exact distribution and its Laplace transform for the waiting time T until the first appearance of an $$S_{i}$$ S i -run of length $$k_{i}$$ k i , for $$i=1,2,\ldots ,r$$ i = 1 , 2 , … , r (whichever comes first), are provided. The exact distribution of the length $$L_{t}$$ L t of the longest run of successes in the time interval [0, t] is also derived. Further, the asymptotic distributions of T and $$L_{t}$$ L t are obtained under general assumptions. Finally, numerical examples and applications in reliability theory, quality control and hypothesis testing are presented.

Suggested Citation

  • Eutichia Vaggelatou, 2024. "On the Longest Run and the Waiting Time for the First Run in a Continuous Time Multi-State Markov Chain," Methodology and Computing in Applied Probability, Springer, vol. 26(4), pages 1-26, December.
    • Handle: RePEc:spr:metcap:v:26:y:2024:i:4:d:10.1007_s11009-024-10127-5
    DOI: 10.1007/s11009-024-10127-5
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    References listed on IDEAS

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    1. Spiros D. Dafnis & Frosso S. Makri, 2022. "Weak runs in sequences of binary trials," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 85(5), pages 573-603, July.
    2. Demetrios Antzoulakos, 1999. "On Waiting Time Problems Associated with Runs in Markov Dependent Trials," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 51(2), pages 323-330, June.
    3. Sigeo Aki, 1992. "Waiting time problems for a sequence of discrete random variables," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 44(2), pages 363-378, June.
    4. Yong Kong, 2017. "The mth longest runs of multivariate random sequences," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(3), pages 497-512, June.
    5. Ebneshahrashoob, M. & Sobel, Milton, 1990. "Sooner and later waiting time problems for Bernoulli trials: frequency and run quotas," Statistics & Probability Letters, Elsevier, vol. 9(1), pages 5-11, January.
    6. Eryilmaz, Serkan & Gong, Min & Xie, Min, 2016. "Generalized sooner waiting time problems in a sequence of trinary trials," Statistics & Probability Letters, Elsevier, vol. 115(C), pages 70-78.
    7. M. Koutras & V. Alexandrou, 1995. "Runs, scans and URN model distributions: A unified Markov chain approach," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 47(4), pages 743-766, December.
    8. Sigeo Aki & Katuomi Hirano, 2007. "On the Waiting Time for the First Success Run," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 59(3), pages 597-602, September.
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