This work is concerned with the asymptotic properties of a singular perturbed nonstationary finite state Markov chain. In a recent paper of the authors, it was shown that as the fluctuation rate of the Markov chain goes to $\infty$, the probability distribution of the Markov chain converges to its time-dependent quasi-equilibrium distribution. In addition, asymptotic expansion of the probability distribution was obtained. This paper is a continuation of our effort in this direction. Upon using the asymptotic expansion, a suitably scaled sequence is examined in detail. Asymptotic normality is obtained. It is shown that the accumulated difference between the indicator process and the quasi-equilibrium distribution converges to a Gaussian process with zero mean. An explicit formula for the covariance function of the Gaussian process is obtained, which depends crucially on the asymptotic expansion.
"A central limit theorem for singularly perturbed nonstationary finite state Markov chains." Ann. Appl. Probab. 6 (2) 650 - 670, May 1996. https://doi.org/10.1214/aoap/1034968148