Optimal Hoeffding bounds for discrete reversible Markov chains

We build optimal exponential bounds for the probabilities of large deviations of sums .k=1nf(Xk) where (Xk) is a finite reversible Markov chain and f is an arbitrary bounded function. These bounds depend only on the stationary mean E.f, the end-points of the support of f, the sample size n and the second largest eigenvalue . of the transition matrix.