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

翻译：我们研究用于数值积分的马尔可夫链蒙特卡洛（MCMC）估计量，其针对的函数无需关于不变分布平方可积。对于具有谱间隙的马尔可夫链，我们证明：对于$p \in (1,2)$的$L^p$函数，绝对均值误差的衰减速度为$n^{1/p -1}$，该速率已知为最优。这一结果改进了当前已知结论——后者需要引入额外参数$\delta>0$，且收敛阶为$n^{(1+\delta)/p-1}$。