We present a version of the Wentzell-Freidlin theory for Markov chains which includes random perturbations not only of deterministic motions but also of Markov chains. Some results for the continuous-time case are obtained as corollaries. In particular, by this method one can treat random perturbations of degenerate diffusions even when the large deviations principle fails.
Yuri Kifer. "A Discrete-Time Version of the Wentzell-Friedlin Theory." Ann. Probab. 18 (4) 1676 - 1692, October, 1990. https://doi.org/10.1214/aop/1176990641