Singularly Perturbed Markov Chains: Convergence and Aggregation

Asymptotic properties of singularly perturbed Markov chains having measurable and/or continuous generators are developed in this work. The Markov chain under consideration has a finite-state space and is allowed to be nonstationary. Its generator consists of a rapidly varying part and a slowly changing part. The primary concerns are on the properties of the probability vectors and an aggregated process that depend on the characteristics of the fast varying part of the generators. The fast changing part of the generators can either consist of l recurrent classes, or include also transient states in addition to the recurrent classes. The case of inclusion of transient states is examined in detail. Convergence of the probability vectors under the weak topology of L² is obtained first. Then under slightly stronger conditions, it is shown that the convergence also takes place pointwise. Moreover, convergence under the norm topology of L² is derived. Furthermore, a process with aggregated states is obtained which converges to a Markov chain in distribution.