Fast computation of expectations and integrals by Markov chain Monte Carlo methods; error bounds and burn-in

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We study fast algorithms for the numerical computation of weighted sums and integrals over domainsthat are high dimensional or have a complicated structure. We want to tune the algorithms in an optimal way, depending on properties of the integrand, the density, and the domain. Weighted sums and integrals are met in numerous applications, especially in statistical physics, in statistics, in financial mathematics and in computer science. It is amazing that the Metropolis algorithm is one of the most widely used algorithms but, nevertheless, not many error bounds are known. For the applications, stopping criteria as well as the time for warming up are most important. Because of the lack of (nonasymptotic) error bounds, these important parts of the algorithms are usually chosen heuristically. In contrast, we proved explicit error bounds. Now we will study further applications of these bounds as well as modified error bounds and algorithms.