Optimal convergence rates of MCMC integration for functions with unbounded second moment

Abstract: We study the Markov chain Monte Carlo (MCMC) estimator for numerical integration for functions that do not need to be square integrable w.r.t. the invariant distribution. For chains with a spectral gap we show that the absolute mean error for $L^p$ functions, with $p \in (1,2)$, decreases like $n^{1/p -1}$, which is known to be the optimal rate. This improves currently known results where an additional parameter $\delta>0$ appears and the convergence is of order $n^{(1+\delta)/p-1}$.