Error bounds of MCMC for functions with unbounded stationary variance

Abstract: We prove explicit error bounds for Markov chain Monte Carlo (MCMC) methods to compute expectations of functions with unbounded stationary variance. We assume that there is a $p\in(1,2)$ so that the functions have finite $L_p$-norm. For uniformly ergodic Markov chains we obtain error bounds with the optimal order of convergence $n^{1/p-1}$ and if there exists a spectral gap we almost get the optimal order. Further, a burn-in period is taken into account and a recipe for choosing the burn-in is provided.