Formulae and recursions for the joint distributions of success runs of several lengths in a two-state Markov chain

Let X1, X2,...,Xn be a time-homogeneous {0, 1}-valued Markov chain. Let Y = (Y1,...,Yr) denote the r-dimensional random vector, where Yi(i = l,...,r) represents the number of success runs of length ki(i = 1,...,r), k = (kl,...,kr). In this paper we obtain exact and recurrence formulae for the probability functions and the probability generating functions of Y, based on four different ways of counting numbers of success runs (i.e. overlapping success runs, non-overlapping runs, the runs with a specified length k or more and the runs with just specified length k). Using our methods, we can obtain the probability function and probability generating function of Y, where Y1,...,Yr be counted numbers by different ways.