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Computing waiting time probabilities related to $$ (k_{1},k_{2},\ldots ,k_{l})$$ ( k 1 , k 2 , … , k l ) pattern

Author

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  • Stathis Chadjiconstantinidis

    (University of Piraeus)

  • Serkan Eryilmaz

    (Atilim University)

Abstract

For a sequence of multi-state trials with l possible outcomes denoted by $$ \left\{ 1,2,\ldots ,l\right\} $$ 1 , 2 , … , l , let E be the event that at least $$k_{1}$$ k 1 consecutive 1s followed by at least $$k_{2}$$ k 2 consecutive 2s,..., followed by at least $$k_{l}$$ k l consecutive ls. Denote by $$T_{r}$$ T r the number of trials for the rth occurrence of the event E in a sequence of multi-state trials. This paper studies the distribution of the waiting time random variable $$T_{r}$$ T r when the sequence consists of independent and identically distributed multi-state trials. In particular, distributional properties of $$ T_{r}$$ T r are examined via matrix-geometric distributions.

Suggested Citation

  • Stathis Chadjiconstantinidis & Serkan Eryilmaz, 2023. "Computing waiting time probabilities related to $$ (k_{1},k_{2},\ldots ,k_{l})$$ ( k 1 , k 2 , … , k l ) pattern," Statistical Papers, Springer, vol. 64(5), pages 1373-1390, October.
    • Handle: RePEc:spr:stpapr:v:64:y:2023:i:5:d:10.1007_s00362-022-01351-7
    DOI: 10.1007/s00362-022-01351-7
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    References listed on IDEAS

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    1. N. Balakrishnan & S. Mohanty & S. Aki, 1997. "Start-Up Demonstration Tests Under Markov Dependence Model with Corrective Actions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 49(1), pages 155-169, March.
    2. Sigeo Aki, 2019. "Waiting time for consecutive repetitions of a pattern and related distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(2), pages 307-325, April.
    3. 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.
    4. Demetrios Antzoulakos & Stathis Chadjiconstantinidis, 2001. "Distributions of Numbers of Success Runs of Fixed Length in Markov Dependent Trials," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(3), pages 599-619, September.
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