On the Asymptotic Normality of Adaptive Multilevel Splitting

TL;DR

This paper proves the consistency and asymptotic normality of the Adaptive Multilevel Splitting method for Markov processes, providing theoretical foundations for its convergence properties.

Contribution

It establishes the first rigorous theoretical results on the convergence and normality of AMS by linking it to Fleming-Viot particle systems.

Findings

Proves consistency of AMS in a general setting.

Establishes asymptotic normality of AMS.

Connects AMS to Fleming-Viot particle systems.

Abstract

Adaptive Multilevel Splitting (AMS for short) is a generic Monte Carlo method for Markov processes that simulates rare events and estimates associated probabilities. Despite its practical efficiency, there are almost no theoretical results on the convergence of this algorithm. The purpose of this paper is to prove both consistency and asymptotic normality results in a general setting. This is done by associating to the original Markov process a level-indexed process, also called a stochastic wave, and by showing that AMS can then be seen as a Fleming-Viot type particle system. This being done, we can finally apply general results on Fleming-Viot particle systems that we have recently obtained.