A Markov-binomial distribution

Let fXi;i 1g denote a sequence of f0;1g-variables and suppose that the sequence forms a Markov Chain. In the paper we study the number of successes Sn = X1 + X2 + ::: + Xn and we study the number of experiments Y (r) up to the r th success. In the i.i.d. case Sn has a binomial distribution and Y (r) has a negative binomial distribution and the asymptotic behaviour is well known. In the more general Markov chain case, we prove a central limit theorem for Sn and provide conditions under which the distribution of Sn can be approximated by a Poisson-type of distribution. We also completely characterize Y (r) and show that Y (r) can be interpreted as the sum of r independent r.v. related to a geometric distribution.