AISM 54, 719-730

© 2002 ISM

## The exact and limiting distributions for the number of successes in success runs within a sequence of Markov-dependent two-state trials

### James C. Fu^{1}, W. Y. Wendy Lou^{2}, Zhi-Dong Bai^{3} and Gang Li^{4}

^{1}Department of Statistics, University of Manitoba,
Winnipeg, Manitoba, Canada R3T 2N2

^{2}Department of Public Health Sciences,
University of Toronto, Toronto, Ontario, Canada M5S 1A8

^{3}Department of Statistics and Applied Probability,
National University of Singapore, Singapore 119260, Singapore

^{4}Biometrics, Organon, Inc., West Orange, NJ
07052, U.S.A.

(Received April 16, 2001; revised October 26, 2001)

Abstract.
The total number of successes in success runs of length
greater than or equal to $k$ in a sequence of $n$ two-state trials is a
statistic that has been broadly used in statistics and probability.
For Bernoulli trials with $k$ equal to one, this statistic has been
shown to have binomial and normal distributions as exact and limiting
distributions, respectively. For the case of Markov-dependent
two-state trials with $k$ greater than one, its exact and limiting
distributions have never been considered in the literature. In this
article, the finite Markov chain imbedding technique and the invariance
principle are used to obtain, in general, the exact and limiting
distributions of this statistic under Markov dependence, respectively.
Numerical examples are given to illustrate the theoretical results.

Key words and phrases:
Finite Markov chain imbedding, transition probability matrix, runs and patterns.

**Source**
(TeX , DVI )