Large deviations for Markov chains and spectrum of matrices

Abstract

This paper deals with large deviations and related problems for Markov chains with countable state spaces. By comparing with some dominating chains that satisfy the large deviation principle, we weaken the requirement for irreducibility to get the large deviation principle. In this way we can also characterize the stationary distributions of the chains via the large deviation rate functions. In particular, we prove that any finite Markov chain satisfies the large deviation principle. The global and local minimums ofthe rate function are shown to be closely related to the principal eigenvalues and the corresponding left and right eigenfunctions of the transition matrix.