On the soonest and latest waiting time distributions: succession quotas

Let be a sequence of independent and identically distributed random variables with let k i . be a given positive integer. We define the random variable N i to be the smallest integer so that a run in i's of length k i has occurred in the subsequence . Then N i 's are correlated variables. Also each N N i has a geometric distribution of order k N i (see Philippou et al. (1983)). The main objective of the present paper is to study the probability generating functions and hence the means and variances of the minimum and the maximum of these m correlated geometric distributions of different orders. It generalizes some of the earlier results in Ebneshahrashoob and Sobel (1990) and Ling (1990). As by products, some recurrence relations among characteristics of order statistics defined on an arbitrary set of random variables are also established.