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The occurrence of sequence patterns in repeated experiments and hitting times in a Markov chain

Author

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  • Gerber, Hans U.
  • Li, Shuo-Yen Robert

Abstract

A martingale argument is used to derive the generating function of the number of i.i.d. experiments it takes to observe a given string of outcomes for the first time. Then, a more general problem can be studied: How many trials does it take to observe a member of a finite set of strings for the first time? It is shown how the answer can be obtained within the framework of hitting times in a Markov chain. For these, a result of independent interest is derived.

Suggested Citation

  • Gerber, Hans U. & Li, Shuo-Yen Robert, 1981. "The occurrence of sequence patterns in repeated experiments and hitting times in a Markov chain," Stochastic Processes and their Applications, Elsevier, vol. 11(1), pages 101-108, March.
    • Handle: RePEc:eee:spapps:v:11:y:1981:i:1:p:101-108
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    Citations

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    Cited by:

    1. Bara Kim & Jeongsim Kim & Jerim Kim, 2020. "Waiting Time Problems for Patterns in a Sequence of Multi-State Trials," Mathematics, MDPI, vol. 8(11), pages 1-16, October.
    2. Ourania Chryssaphinou & Stavros Papastavridis & Eutichia Vaggelatou, 1999. "On the Number of Appearances of a Word in a Sequence of I.I.D. Trials," Methodology and Computing in Applied Probability, Springer, vol. 1(3), pages 329-348, October.
    3. Fisher, Evan & Cui, Shiliang, 2010. "Patterns generated by -order Markov chains," Statistics & Probability Letters, Elsevier, vol. 80(15-16), pages 1157-1166, August.
    4. Masayuki Uchida, 1998. "On Generating Functions of Waiting Time Problems for Sequence Patterns of Discrete Random Variables," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 50(4), pages 655-671, December.
    5. Pozdnyakov, Vladimir, 2008. "On occurrence of patterns in Markov chains: Method of gambling teams," Statistics & Probability Letters, Elsevier, vol. 78(16), pages 2762-2767, November.
    6. Zajkowski, Krzysztof, 2014. "A note on the gambling team method," Statistics & Probability Letters, Elsevier, vol. 85(C), pages 45-50.
    7. Sharma, Dheeraj & Talpur, Mir Ghulam Haider, 2016. "An Examination of One Dimension Marginal Distributions: Selling and Non-selling Activities of a Salesperson," IIMA Working Papers WP2016-03-48, Indian Institute of Management Ahmedabad, Research and Publication Department.

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