Optimal Hoeffding bounds for discrete reversible Markov chains

Abstract

We build optimal exponential bounds for the probabilities of large deviations of sums ∑_k=1ⁿf(X_k) where (X_k) is a finite reversible Markov chain and f is an arbitrary bounded function. These bounds depend only on the stationary mean ${\mathbb {E}}_{\pi}f,$ the end-points of the support of f, the sample size n and the second largest eigenvalue λ of the transition matrix.