Rates of Convergence and CLTs for Subcanonical Debiased MLMC

Abstract

In constructing debiased multi-level Monte Carlo (MLMC) estimators, one must choose a randomization distribution. In some algorithmic contexts, an optimal choice for the randomization distribution leads to a setting in which the mean time to generate an unbiased observation is infinite. This paper extends the well known efficiency theory for Monte Carlo algorithms in the setting of a finite mean for this generation time to the infinite mean case. The theory draws upon stable law weak convergence results, and leads directly to exact convergence rates and central limit theorems (CLTs) for various debiased MLMC algorithms, most particularly as they arise in the context of stochastic differential equations. Our CLT theory also allows simulators to construct asymptotically valid confidence intervals for such infinite mean MLMC algorithms.