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

Part of: Probabilistic methods, simulation and stochastic differential equations Markov processes

Published online by Cambridge University Press:  14 February 2025

Julian Hofstadler Open the ORCID record for Julian Hofstadler [Opens in a new window]
Julian Hofstadler*
Affiliation:
University of Passau
*
*Postal address: Faculty of Computer Science and Mathematics, University of Passau, Innstraße 33, 94032 Passau, Germany. Email: [email protected]
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Abstract

We study the Markov chain Monte Carlo estimator for numerical integration for functions that do not need to be square integrable with respect to 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 $\delta \gt 0$ appears and the convergence is of order $n^{(({1+\delta})/{p})-1}$.

Keywords

Markov chain Monte Carlospectral gapabsolute mean error

MSC classification

Primary: 65C05: Monte Carlo methods 60J22: Computational methods in Markov chains
Secondary: 65C20: Models, numerical methods
Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 7
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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