Annals of the Institute of Statistical Mathematics

, Volume 65, Issue 3, pp 513–527

Recursive equations in finite Markov chain imbedding


  • Yu-Fei Hsieh
    • Department of StatisticsUniversity of Manitoba
    • Department of StatisticsUniversity of Manitoba

DOI: 10.1007/s10463-012-0381-x

Cite this article as:
Hsieh, Y. & Wu, T. Ann Inst Stat Math (2013) 65: 513. doi:10.1007/s10463-012-0381-x


In this paper, recursive equations for waiting time distributions of r-th occurrence of a compound pattern are studied via the finite Markov chain imbedding technique under overlapping and non-overlapping counting schemes in sequences of independent and identically distributed (i.i.d.) or Markov dependent multi-state trials. Using the relationship between number of patterns and r-th waiting time, distributions of number of patterns can also be obtained. The probability generating functions are also obtained. Examples and numerical results are given to illustrate our theoretical results.


Recursive equation Simple and compound patterns Waiting time Finite Markov chain imbedding Probability generating function

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© The Institute of Statistical Mathematics, Tokyo 2012