Error bounds for computing the expectation by Markov chain Monte Carlo

We study the error of reversible Markov chain Monte Carlo methods for approximating the expectation of a function. Explicit error bounds with respect to different norms of the function are proven. By the estimation the well known asymptotical limit of the error is attained, i.e. there is no gap between the estimate and the asymptotical behavior. We discuss the dependence of the error on a burn-in of the Markov chain. Furthermore we suggest and justify a specific burn-in for optimizing the algorithm.