Asymptotic Normality and Exponential Bounds

Abstract

This chapter examines the asymptotic properties of the singularly perturbed Markov chains α^ε(.) under weak irreducibility from another angle. The central theme here is on limit results of unscaled as well as scaled sequences of occupation measures, which include law of large numbers for an unscaled sequence, exponential upper bounds, and asymptotic normality of a suitably scaled sequence of occupation times. The study in Chapter 4 is on the probability distribution of α^ε(.) through the corresponding forward equation and is mainly a deterministic approach, whereas the current chapter is largely probabilistic in nature. It further exploits the deviation of the functional occupation times from its quasi-stationary distribution. We show the centered deviation having zero limit in probability, prove the convergence of a properly scaled and centered sequence of occupation times, characterize the limit process by deriving explicit formulae for the mean and covariance functions, and provide exponential bounds for the normalized process. It is worthwhile to note that the limit covariance function depends on the boundary layer terms, which is different from most of the existing results of central limit type.