# Recursive equations in finite Markov chain imbedding

Article

- First Online:

- Received:
- Revised:

DOI: 10.1007/s10463-012-0381-x

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

- 169 Downloads

## Abstract

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.

### Keywords

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

© The Institute of Statistical Mathematics, Tokyo 2012