On approximation of Markov binomial distributions

For a Markov chain X = {Xi, i = 1, 2, . . . , n} with the state space {0, 1}, the random variable S := ∑n i=1 Xi is said to follow a Markov binomial distribution. The exact distribution of S, denoted as LS, is very computationally intensive for large n [see Gabriel (1959) and Bhat and Lal (1988)] and this paper concerns suitable approximate distributions for LS when X is stationary. We conclude that the negative binomial and binomial distributions are appropriate approximations for LS depending on VarS greater than and less than ES respectively. Also, due to the unique structure of the distribution, we are able to derive explicit error estimates for these approximations.