×

zbMATH — the first resource for mathematics

Oscillation properties of expected stopping times and stopping probabilities for patterns consisting of consecutive states in Markov chains. (English) Zbl 1456.60179
Summary: We investigate a Markov chain with a state space \(1, 2, \ldots, r\) stopping at appearance of patterns consisting of two consecutive states. It is observed that the expected stopping times of the chain have surprising oscillating dependencies on starting positions. Analogously, the stopping probabilities also have oscillating dependencies on terminal states. In a nonstopping Markov chain the frequencies of appearances of two consecutive states are found explicitly.
MSC:
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
05C81 Random walks on graphs
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] G. Blom and D. Thorburn, “How many random digits are required until given sequences are obtained?”, J. Appl. Probab. 19:3 (1982), 518-531. Mathematical Reviews (MathSciNet): MR664837
Digital Object Identifier: doi:10.2307/3213511
· Zbl 0493.60071
[2] E. Fisher and S. Cui, “Patterns generated by \(m\) th-order Markov chains”, Statist. Probab. Lett. 80:15-16 (2010), 1157-1166. Mathematical Reviews (MathSciNet): MR2657478
Digital Object Identifier: doi:10.1016/j.spl.2010.03.011
· Zbl 1200.60061
[3] J. C. Fu and Y. M. Chang, “On probability generating functions for waiting time distributions of compound patterns in a sequence of multistate trials”, J. Appl. Probab. 39:1 (2002), 70-80. Mathematical Reviews (MathSciNet): MR1895144
Zentralblatt MATH: 1008.60031
Digital Object Identifier: doi:10.1239/jap/1019737988
Project Euclid: euclid.jap/1019737988
· Zbl 1008.60031
[4] R. J. Gava and D. Salotti, “Stopping probabilities for patterns in Markov chains”, J. Appl. Probab. 51:1 (2014), 287-292. Mathematical Reviews (MathSciNet): MR3189458
Zentralblatt MATH: 1291.60147
Digital Object Identifier: doi:10.1239/jap/1395771430
Project Euclid: euclid.jap/1395771430
· Zbl 1291.60147
[5] H. U. Gerber and S.-Y. R. Li, “The occurrence of sequence patterns in repeated experiments and hitting times in a Markov chain”, Stochastic Process. Appl. 11:1 (1981), 101-108. Mathematical Reviews (MathSciNet): MR608011
Digital Object Identifier: doi:10.1016/0304-4149(81)90025-9
· Zbl 0449.60050
[6] J. Glaz, M. Kulldorff, V. Pozdnyakov, and J. M. Steele, “Gambling teams and waiting times for patterns in two-state Markov chains”, J. Appl. Probab. 43:1 (2006), 127-140. Mathematical Reviews (MathSciNet): MR2225055
Zentralblatt MATH: 1105.60051
Digital Object Identifier: doi:10.1239/jap/1143936248
Project Euclid: euclid.jap/1143936248
· Zbl 1105.60051
[7] L. J. Guibas and A. M. Odlyzko, “String overlaps, pattern matching, and nontransitive games”, J. Combin. Theory Ser. A 30:2 (1981), 183-208. Mathematical Reviews (MathSciNet): MR611250
Digital Object Identifier: doi:10.1016/0097-3165(81)90005-4
· Zbl 0454.68109
[8] Q. Han and S. Aki, “Waiting time problems in a two-state Markov chain”, Ann. Inst. Statist. Math. 52:4 (2000), 778-789. Mathematical Reviews (MathSciNet): MR1820750
Zentralblatt MATH: 0989.60018
Digital Object Identifier: doi:10.1023/A:1017537629251
· Zbl 0989.60018
[9] S.-Y. R. Li, “A martingale approach to the study of occurrence of sequence patterns in repeated experiments”, Ann. Probab. 8:6 (1980), 1171-1176. Mathematical Reviews (MathSciNet): MR602390
Digital Object Identifier: doi:10.1214/aop/1176994578
Project Euclid: euclid.aop/1176994578
· Zbl 0447.60006
[10] L. Lovász, “Random walks on graphs: a survey”, pp. 353-397 in Combinatorics, Paul Erdős is eighty (Keszthely, 1993), Bolyai Soc. Math. Stud. 2, János Bolyai Math. Soc., Budapest, 1996. Mathematical Reviews (MathSciNet): MR1395866
[11] V. · Zbl 1151.60327
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.