Abstract
For a Markov chain $\mathbf{X}=\{X_i, i=1, 2, …, n\}$ with the state space $\{0, 1\}$, the random variable $S:=∑_{i=1}^nX_i$ 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 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.
Citation
Aihua Xia. Mei Zhang. "On approximation of Markov binomial distributions." Bernoulli 15 (4) 1335 - 1350, November 2009. https://doi.org/10.3150/09-BEJ194
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