We consider adaptive increasingly rare Markov chain Monte Carlo (AIR MCMC), which is an adaptive MCMC method, where the adaptation concerning the past happens less and less frequently over time. Under a contraction assumption for a Wasserstein-like function we deduce upper bounds of the convergence rate of Monte Carlo sums taking a renormalisation factor into account that is close to the one that appears in a law of the iterated logarithm. We demonstrate the applicability of our results by considering different settings, among which are those of simultaneous geometric and uniform ergodicity. All proofs are carried out on an augmented state space, including the classical non-augmented setting as a special case. In contrast to other adaptive MCMC limit theory, some technical assumptions, like diminishing adaptation, are not needed.

翻译：我们考虑自适应递增稀有马尔可夫链蒙特卡洛方法（AIR MCMC），这是一种自适应MCMC方法，其中关于过去历史的自适应随时间推移发生得越来越不频繁。在Wasserstein类函数的收缩假设下，我们推导出蒙特卡洛和的收敛速率上界，该上界考虑了接近迭代对数律中出现的重整化因子。我们通过考虑不同设置（包括同时满足几何与一致遍历性的情形）展示了结果的适用性。所有证明均在增广状态空间上进行，而经典的非增广设置作为特例被包含其中。与其他自适应MCMC极限理论相比，本方法无需诸如递减自适应等技术性假设。