Central limit theorems for multilevel Monte Carlo methods

TL;DR

This paper establishes broader conditions under which central limit theorems hold for multilevel Monte Carlo estimators, introducing a new estimator to ensure CLT applicability in more cases.

Contribution

It shows that uniform integrability is not necessary for CLT in MLMC and proposes a mass-shifted estimator that guarantees CLT under weaker conditions.

Findings

CLT applies when variance decay dominates cost rate (β > γ).

Mass-shifted MLMC estimator ensures CLT in all settings.

Cost increase for mass-shifted estimator is at most logarithmic in accuracy.

Abstract

In this work, we show that uniform integrability is not a necessary condition for central limit theorems (CLT) to hold for normalized multilevel Monte Carlo (MLMC) estimators and we provide near optimal weaker conditions under which the CLT is achieved. In particular, if the variance decay rate dominates the computational cost rate (i.e., $\beta > \gamma$), we prove that the CLT applies to the standard (variance minimizing) MLMC estimator. For other settings where the CLT may not apply to the standard MLMC estimator, we propose an alternative estimator, called the mass-shifted MLMC estimator, to which the CLT always applies. This comes at a small efficiency loss: the computational cost of achieving mean square approximation error $\mathcal{O}(\epsilon^2)$ is at worst a factor $\mathcal{O}(\log(1/\epsilon))$ higher with the mass-shifted estimator than with the standard one.