Extended Laplace principle for empirical measures of a Markov chain

Abstract

We consider discrete-time Markov chains with Polish state space. The large deviations principle for empirical measures of a Markov chain can equivalently be stated in Laplace principle form, which builds on the convex dual pair of relative entropy (or Kullback– Leibler divergence) and cumulant generating functional f ↦ ln ʃ exp (f). Following the approach by Lacker (2016) in the independent and identically distributed case, we generalize the Laplace principle to a greater class of convex dual pairs. We present in depth one application arising from this extension, which includes large deviation results and a weak law of large numbers for certain robust Markov chains—similar to Markov set chains—where we model robustness via the first Wasserstein distance. The setting and proof of the extended Laplace principle are based on the weak convergence approach to large deviations by Dupuis and Ellis (2011).