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Optimal convergence rates of MCMC integration for functions with unbounded second moment
- Part of
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- Journal:
- Journal of Applied Probability , First View
- Published online by Cambridge University Press:
- 14 February 2025, pp. 1-7
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- Article
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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 
$n^{(({1+\delta})/{p})-1}$.
