A Markov-binomial distribution

Abstract

Let {Xi, 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 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.