ON NUMBER OF OCCURRENCES OF SUCCESS RUNS OF SPECIFIED LENGTH IN A HIGHER-ORDER TWO-STATE MARKOV CHAIN

MASAYUKI UCHIDA

The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106-8569, Japan

(Received October 24, 1996; revised August 14, 1997)

Abstract. Let X_{−m+1}, X_{−m+2}, …, X_{0}, X_{1}, X_{2}, …, X_{n} be a time-homogene-ous {0, 1}-valued m-th order Markov chain. The probability distributions of numbers of runs of “1” of length k (k ≥ m) and of “1” of length k (k < m) in the sequence of a {0, 1}-valued m-th order Markov chain are studied. There are some ways of counting numbers of runs with length k. This paper studies the distributions based on four ways of counting numbers of runs, i.e., the number of non-overlapping runs of length k, the number of runs with length greater than or equal to k, the number of overlapping runs of length k and the number of runs of length exactly k.

Key words and phrases: Probability generating function, discrete distribution, binomial distribution, binomial distribution of order k, higher order Markov chain.