Entropy and Large Deviations for Discrete-Time Markov Chains

Let $E$ be a denumerable state space and $X$ be a homogeneous Markov chain on $E$ with kernel $P$. Then the chain $X$ verifies a weak Sanov's theorem, i.e., a weak large deviation principle holds for the law of the pair empirical measure. This LDP holds for any discrete state space Markov chain, not necessarily ergodic or irreducible. It is also known that a strong LDP cannot hold in the present framework. The result is obtained by a new method, which allows us to extend the LDP from a finite state space setting to a denumerable one, somehow like the projective limit approach. The analysis presented here offers some by-products, among which there are a contraction principle for the weak LDP, leading directly to a weak Sanov's theorem for the one-dimensional empirical measure. A refined analysis of the ubiquitous entropy function $H$ proves to be useful in other frameworks, e.g., continuous time or stochastic networks, and allows us to improve the sharpness of asymptotics.