Abstract
Suppose that C is a finite collection of patterns. Observe a Markov chain until one of the patterns in C occurs as a run. This time is denoted by τ. In this paper, we aim to give an easy way to calculate the mean waiting time E(τ ) and the stop** probabilities P(τ = τ A ) with A ∈ C, where τ A is the waiting time until the pattern A appears as a run.
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References
J Brofos. A Markov chain analysis of a pattern matching coin game, ar**v preprint, ar**v: 1406.2212, 2014.
O Chrysaphinou, S Papastavridis. The occurrence of sequence patterns in repeated dependent experiments, Theory Probab Appl, 1991, 35(1): 145–152.
JC Fu, YM Chang. On probability generating functions for waiting time distributions of compound patterns in a sequence of multistate trials, J Appl Probab, 2002, 39: 70–80.
RJ Gava, D Salotti. Stop** probabilities for patterns in Markov chains, J Appl Probab, 2014, 51(1): 287–292.
HU Gerber, SR Li. The occurrence of sequence patterns in repeated experiments and hitting times in a Markov chain, Stochastic Process Appl, 1981, 11(1): 101–108.
J Glaz, M Kulldorff, V Pozdnyakov, JM Steele. Gambling teams and waiting times for patterns in two-state Markov chains, J Appl Probab, 2006, 43(1): 127–140.
LJ Guibas, AM Odlyzko. String overlaps, pattern matching, and nontransitive games, J Combin Theory Ser A, 1981, 30(2): 183–208.
SR Li. A martingale approach to the study of occurrence of sequence patterns in repeated experiments, Ann Probab, 1980, 8(6): 1171–1176.
JI Naus. The distribution of the size of the maximum cluster of points on a line, J Amer Statist Assoc, 1965, 60(310): 532–538.
J I Naus, VT Stefanov. Double-scan statistics, Methodol Comput Appl Probab, 2002, 4(2): 163–180.
Y Nishiyama. Pattern matching probabilities and paradoxes as a new variation on Penney’s coin game, Int J Pure Appl Math, 2010, 59(3): 357–366.
V Pozdnyakov. On occurrence of patterns in Markov chains: method of gambling teams, Statist Probab Lett, 2008, 78(16): 2762–2767.
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The authors wish to express their thanks to the referee for his/her constructive suggestions and many helpful comments.
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Supported by the National Natural Science Foundation of China (11771286, 11371317) and by the Zhejiang Provincial Natural Science Foundation of China (LQ18A010007).
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Zhao, Mz., Xu, D. & Zhang, Hz. Waiting times and stop** probabilities for patterns in Markov chains. Appl. Math. J. Chin. Univ. 33, 25–34 (2018). https://doi.org/10.1007/s11766-018-3522-z
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DOI: https://doi.org/10.1007/s11766-018-3522-z