Waiting times and stopping probabilities for patterns in Markov chains

  • Min-zhi Zhao
  • Dong Xu
  • Hui-zeng Zhang


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 stopping probabilities P(τ = τ A ) with AC, where τ A is the waiting time until the pattern A appears as a run.


pattern Markov chain stopping probability waiting time 

MR Subject Classification

60J10 60J22 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors wish to express their thanks to the referee for his/her constructive suggestions and many helpful comments.


  1. [1]
    J Brofos. A Markov chain analysis of a pattern matching coin game, arXiv preprint, arXiv: 1406.2212, 2014.Google Scholar
  2. [2]
    O Chrysaphinou, S Papastavridis. The occurrence of sequence patterns in repeated dependent experiments, Theory Probab Appl, 1991, 35(1): 145–152.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    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.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    RJ Gava, D Salotti. Stopping probabilities for patterns in Markov chains, J Appl Probab, 2014, 51(1): 287–292.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    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.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    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.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    LJ Guibas, AM Odlyzko. String overlaps, pattern matching, and nontransitive games, J Combin Theory Ser A, 1981, 30(2): 183–208.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    SR Li. A martingale approach to the study of occurrence of sequence patterns in repeated experiments, Ann Probab, 1980, 8(6): 1171–1176.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    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.MathSciNetCrossRefGoogle Scholar
  10. [10]
    J I Naus, VT Stefanov. Double-scan statistics, Methodol Comput Appl Probab, 2002, 4(2): 163–180.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    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.MathSciNetzbMATHGoogle Scholar
  12. [12]
    V Pozdnyakov. On occurrence of patterns in Markov chains: method of gambling teams, Statist Probab Lett, 2008, 78(16): 2762–2767.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Editorial Committee of Applied Mathematics-A Journal of Chinese Universities and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesZhejiang UniversityHangzhouChina
  2. 2.Department of MathematicsHangzhou Normal UniversityHangzhouChina

Personalised recommendations