A unified approach to binomial-type distributions of order k

In this paper, we consider the distribution of the number of "1"-runs of length k in a sequence of {0,1}-valued random variables of length n by using a new (unified) counting scheme called l-overlapping counting. Here, k and n are positive integers with k ≦ and l is an integer less than k. We obtain the prohabi!ity generating function of the distribution of the number of eoverlapping "in-runs of iength k in the sequence, even when the underiying sequence is a dependent sequence such as a highcr order Markov chaic.