Distributions related to events

Let Z 1 , Z 2 , … be a sequence of Bernoulli trials with success probability p = Pr ( Z t = 1 ) and failure probability q = Pr ( Z t = 0 ) = 1 − p , t ⩾ 1 . For positive integers k 1 and k 2 we consider the events E 1 : at least k 1 consecutive 0ʹs are followed by at least k 2 consecutive 1ʹs, E 2 : exactly k 1 consecutive 0ʹs are followed by exactly k 2 consecutive 1ʹs and E 3 : at most k 1 consecutive 0ʹs are followed by at most k 2 consecutive 1ʹs. Denote by X n ( i ) the number of occurrences of the event E i ( i = 1 , 2 , 3 ) in Z 1 , Z 2 , … , Z n ( n ⩾ 1 ) , and let T r ( i ) be the waiting time for the r-th occurrence of the event E i ( i = 1 , 2 , 3 ) in Z 1 , Z 2 , … . 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 ) of X n ( i ) and T r ( i ) ( i = 1 , 2 , 3 ) . An application is also given.