Approximation for counts of 2-runs in a two state Markov chain

We consider a Markov chain X = {X _i , i = 1, 2, ...} with the state space {0, 1}, and define W = Σ _i=1 ⁿ X _i X _i+1, which is the number of 2-runs in X before time n+1. In this paper, we prove that the negative binomial distribution is an appropriate approximation for $\mathcal{L}W$ when VarW is greater than $\mathbb{E}W$ . The error estimate obtained herein improves the corresponding result in previous literatures.