Optimal Chernoff and Hoeffding Bounds for Finite Markov Chains

This paper develops an optimal Chernoff type bound for the probabilities of large deviations of sums $\sum_{k=1}^n f (X_k)$ where $f$ is a real-valued function and $(X_k)_{k \in \mathbb{N}_0}$ is a finite Markov chain with an arbitrary initial distribution and an irreducible stochastic matrix coming from a large class of stochastic matrices. Our bound is optimal in the large deviations sense attaining a constant prefactor and an exponential decay with the optimal large deviations rate. Moreover through a Pinsker type inequality and a Hoeffding type lemma, we are able to loosen up our Chernoff type bound to a Hoeffding type bound and reveal the sub-Gaussian nature of the sums. Finally we show a uniform multiplicative ergodic theorem for our class of Markov chains.