On approximation of Markov binomial distributions

For a Markov chain $\mathbf{X}=\{X_i,i=1,2,\mathrm{.}\mathrm{.}\mathrm{.},n\}$ with the state space $\{0,1\}$, the random variable $S:={\sum }_{i=1}^n{X}_i$ is said to follow a Markov binomial distribution. The exact distribution of $S$, denoted $\mathcal{L}S$, is very computationally intensive for large $n$ (see Gabriel [Biometrika 46 (1959) 454–460] and Bhat and Lal [Adv. in Appl. Probab. 20 (1988) 677–680]) and this paper concerns suitable approximate distributions for $\mathcal{L}S$ when $\mathbf{X}$ is stationary. We conclude that the negative binomial and binomial distributions are appropriate approximations for $\mathcal{L}S$ when $\mathrm{Var}S$ is greater than and less than $\mathbb{E}S$, respectively. Also, due to the unique structure of the distribution, we are able to derive explicit error estimates for these approximations.