Sequences of random matrices modulated by a discrete-time Markov chain*

Abstract

In this work, we consider a number of matrix-valued random sequences that are modulated by a discrete-time Markov chain having a finite space. Assuming that the state space of the Markov chain is large, our main effort in this paper is devoted to reducing the computation complexity. To achieve this goal, our formulation uses time-scale separation of the Markov chain. The state-space of the Markov chain is split into subspaces. Next, the states of the Markov chain in each subspace are aggregated into a “super” state. Then we normalize the matrix-valued sequences that are modulated by the two-time-scale Markov chain. Under simple conditions, we derive a scaling limit of the centered and scaled sequence by using a martingale averaging approach. The study is carried out through a functional. It is shown that the scaled and interpolated sequence converges weakly to a switching diffusion.