JOINT DISTRIBUTIONS OF RUNS IN A SEQUENCE OF MULTI-STATE TRIALS

QING HAN AND SIGEO AKI

Department of Informatics and Mathematical Science, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama-cho, Toyonaka, Osaka 560-8531, Japan

(Received March 10, 1997; revised April 24, 1998)

Abstract. In this paper we introduce a Markov chain imbeddable vector of multinomial type and a Markov chain imbeddable variable of returnable type and discuss some of their properties. These concepts are extensions of the Markov chain imbeddable random variable of binomial type which was introduced and developed by Koutras and Alexandrou (1995, Ann. Inst. Statist. Math., 47, 743–766). By using the results, we obtain the distributions and the probability generating functions of numbers of occurrences of runs of a specified length based on four different ways of counting in a sequence of multi-state trials. Our results also yield the distribution of the waiting time problems.

Key words and phrases: Run statistics, distributions of order k, probability generating function, Markov chain imbedding, multi-state trials, waiting time problem.