A MARKOV-BINOMIAL DISTRIBUTION

Let {X-i , i >= 1} denote a sequence of {0,1}-variables and suppose that the sequence forms a Markov Chain. In the paper we study the number of succ esses Sn = X-1 + X-2 + ... + X-n and we study the number of experiments Y(r) up to th er-th success. In the i.i.d. case S-n 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 S-n and provide conditions under which the distribution of S-n can be approximated by a Poisson-type of distribution. We also comp letely characterize Y(r) and show that Y(r) can be interpreted as the sum of r independent r.v. related to a geometric distribution.