DISTRIBUTIONS OF NUMBERS OF SUCCESS RUNS OF FIXED LENGTH IN MARKOV DEPENDENT TRIALS

DEMETRIOS L. ANTZOULAKOS AND STATHIS CHADJICONSTANTINIDIS

Department of Statistics and Insurance Science, University of Piraeus, 80 Karaoli & Dimitriou Str., 18534 Piraeus, Greece, e-mail: dantz@unipi.gr

(Received January 11, 1999; revised August 9, 1999)

Abstract. Let {Z_{n}, n ≥ 1} be a time-homogeneous {0, 1}-valued Markov chain, and let N_{n} be a random variable denoting the number of runs of “1” of length k in the first n trials. In this article we conduct a systematic study of N_{n} by establishing formulae for the evaluation of its probability generating function, probability mass function and moments. This is done in three different enumeration schemes for counting runs of length k, the “non-overlapping”, the “overlapping” and the “at least” scheme. In the special case of i.i.d. trials several new results are established.

Key words and phrases: Binomial/negative binomial distribution of order k, success runs, Markov chain, probability generating function, probability mass function, moments.