### 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 X1, X2, .... be a sequence of nonnegative integer valued random variables. For each nonnegative integer i, we are given a positive integer ki. For every i =0,1,2, .... , Ei denotes the event that a run of i of length ki occurs in the sequence X1, X2, ..... For the sequence X1, X2, ...., the generalized pgf's of the distributions of the waiting times until the r-th occurrence among the events { Ei }i=0\infty 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.

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