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 $n^{(1+\delta )/p-1}$.

arXiv commentThe title has changed, the proofs have been simplified, and some typos were corrected

References in corpus (7)

• Spectral gaps for a Metropolis-Hastings algorithm in infinite dimensions
• Curvature, concentration and error estimates for Markov chain Monte Carlo
• Explicit error bounds for Markov chain Monte Carlo
• Nonasymptotic bounds on the estimation error of MCMC algorithms
• Explicit convergence bounds for Metropolis Markov chains: isoperimetry, spectral gaps and profiles
• Error bounds of MCMC for functions with unbounded stationary variance
• Spectral gap of nonreversible Markov chains