On the limit of the Markov binomial distribution

Let X 1 X 2, · ·· be a Markov Bernoulli sequence with initial probabilities p of success and q = 1 – p of failure, and probabilities 1 – (1 – π) p, (1 – π) p in the first row and (1 – π) (1 – p), (1 – π) p + πin the second row of the transition matrix. If we define Sn = Σ i=1 n Xi , then the limit distribution P{Sn = k} is obtained when n →∞, np →λ.