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 the l 2-, l 4- and l ∞-norm of the function are proven. By the estimation the well-known asymptotical limit of the error is attained, i.e. our bounds are correct to first order as n → ∞. 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.