PhD Dissertation Defense - Huy Nguyen
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Huy Nguyen, PhD Candidate
Sequences of random matrices modulated by a discrete-time Markov chain
In this dissertation, 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 work is devoted to reducing the 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 limit is considered through a functional. It is shown that the scaled and interpolated sequence converges weakly to a switching diffusion. Towards the end of the work, we also indicate how we may handle matrix-valued processes directly. Certain tail probability estimates are obtained.