WAITING TIME PROBLEMS FOR A SEQUENCE OF DISCRETE RANDOM VARIABLES

SIGEO AKI

Department of Mathematical Science, Faculty of Engineering Science, Osaka University, Machikaneyama-cho, Toyonaka, Osaka 560, Japan

(Received September 20, 1990; revised June 13, 1991)

Abstract. Let X_{1}, X_{2}, … be a sequence of nonnegative integer valued random variables. For each nonnegative integer i, we are given a positive integer k_{i}. For every i = 0, 1, 2, …, E_{i} denotes the event that a run of i of length k_{i} occurs in the sequence X_{1}, X_{2}, . . .. For the sequence X_{1}, X_{2}, …, the generalized pgf’s of the distributions of the waiting times until the r-th occurrence among the events {E_{i}}_{i=0}^{∞} are obtained. Though our situations are general, the results are very simple. For the special cases that X’s are i.i.d. and {0, 1}-valued, the corresponding results are consistent with previously published results.

Key words and phrases: Sooner and later problems, generalized probability generating function, discrete distributions, binary sequence of order k.