On approximation of Markov binomial distributions

Abstract

For a Markov chain X = {Xi, i = 1, 2,⋯, n} with the state space {0, 1}, the random variable S := ∑ni= 1Xi is said to follow a Markov binomial distribution. The exact distribution of 5, denoted £5, is very computationally intensive for large n (see Gabriel [Biometrika 46 (1959) 454-460] and Bhat and LaI [Adv. in Appl. Probab. 20 (1988) 677-680]) and this paper concerns suitable approximate distributions for CS when X is stationary. We conclude that the negative binomial and binomial distributions are appropriate approximations for LS when Var S is greater than and less than ES, respectively. Also, due to the unique structure of the distribution, we are able to derive explicit error estimates ..