Explicit error bounds for lazy reversible Markov chain Monte Carlo

We prove explicit, i.e., non-asymptotic, error bounds for Markov Chain Monte Carlo methods, such as the Metropolis algorithm. The problem is to compute the expectation (or integral) of f$f$ with respect to a measure π$\pi$ which can be given by a density ?$?$ with respect to another measure. A straight simulation of the desired distribution by a random number generator is in general not possible. Thus it is reasonable to use Markov chain sampling with a burn-in. We study such an algorithm and extend the analysis of Lovasz and Simonovits L. Lovász, M. Simonovits, Random walks in a convex body and an improved volume algorithm, Random Structures Algorithms 4 (4) (1993) 359–412] to obtain an explicit error bound.