Distributions related to (k1,k2)(k1,k2) events

Let Z1,Z2,…Z1,Z2,… be a sequence of Bernoulli trials with success probability p=Pr(Zt=1)p=Pr(Zt=1) and failure probability q=Pr(Zt=0)=1−pq=Pr(Zt=0)=1−p, t⩾1t⩾1. For positive integers k1k1 and k2k2 we consider the events E1E1: at least k1k1 consecutive 0's are followed by at least k2k2 consecutive 1's, E2E2: exactly k1k1 consecutive 0's are followed by exactly k2k2 consecutive 1's and E3E3: at most k1k1 consecutive 0's are followed by at most k2k2 consecutive 1's. Denote by Xn(i) the number of occurrences of the event Ei(i=1,2,3) in Z1,Z2,…,Zn(n⩾1), and let Tr(i) be the waiting time for the r  -th occurrence of the event Ei(i=1,2,3) in Z1,Z2,…Z1,Z2,…. In the present paper we employ the Markov chain embedding technique to derive exact formulas for the probability generating functions, the probability mass functions and the m  -th moments (m⩾1)(m⩾1) of Xn(i) and Tr(i)(i=1,2,3). An application is also given.