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}

1Department of Statistics, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2 2Department of Public Health Sciences, University of Toronto, Toronto, Ontario, Canada M5S 1A8

3Department of Statistics and Applied Probability, National University of Singapore,

Singapore 119260, Singapore 4Biometrics, 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.