Singularly perturbed Markov chains: Convergence and aggregation

Asymptotic properties of singularly perturbed Markov chains having measurab le and/or continuous generators are developed in this work. The Markov chai n under consideration has a finite-state space and is allowed to be nonstat ionary. Its generator consists of a rapidly varying part and a slowly chang ing part. The primary concerns are on the properties of the probability vec tors and an aggregated process that depend on the characteristics of the fa st varying part of the generators. The fast changing part of the generators call tither consist of l recurrent classes, or include also transient stal es 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 L-2 is obtained first. Then under slightly stronger c onditions, it is shown that the convergence also takes place pointwise. Mor eover, convergence under the norm topology of L-2 is derived. Furthermore, a process with aggregated states is obtained which converges to a Markov ch ain in distribution. (C) 2000 Academic Press.