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JOINT DISTRIBUTIONS OF RUNS IN A SEQUENCE

OF MULTI-STATE TRIALS

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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.

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