Large deviations for a Markov chain in a random landscape

Let (Sk)k.0 be a Markov chain with state space E and (.x)x.E be a family of .p-valued random vectors assumed independent of the Markov chain. The .x could be assumed independent and identically distributed or could be Gaussian with reasonable correlations. We study the large deviations of the sum https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20180209065921628-0725:S0001867800011617:S0001867800011617_inline1.gif?pub-status=live