Optimal convergence rates of MCMC integration for functions with unbounded second moment
<|reference_start|>Optimal convergence rates of MCMC integration for functions with unbounded second moment: 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 $n^{(1+\delta)/p-1}$.<|reference_end|>
@article{hofstadler2024optimal,
    title={Optimal convergence rates of MCMC integration for functions with
  unbounded second moment},
    author={Julian Hofstadler},
    journal={arXiv preprint arXiv:2403.16920},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.16920},
    primaryClass={math.NA cs.NA}
}