Central Limit Theorem for Adaptative Multilevel Splitting Estimators in an Idealized Setting

TL;DR

This paper proves a central limit theorem for adaptive multilevel splitting estimators in an idealized setting, demonstrating their asymptotic normality and providing numerical evidence of Gaussian-based confidence intervals.

Contribution

It extends previous work by establishing the asymptotic normality of estimators as the system size grows, using functional analysis of characteristic functions.

Findings

Estimators are asymptotically normal as h approaches infinity.

Numerical simulations confirm convergence to Gaussian confidence intervals.

Theoretical analysis relies on solving a functional equation for the characteristic function.

Abstract

The Adaptive Multilevel Splitting algorithm is a very powerful and versatile iterative method to estimate the probability of rare events, based on an interacting particle systems. In an other article, in a so-called idealized setting, the authors prove that some associated estimators are unbiased, for each value of the size n of the systems of replicas and of resampling number k. Here we go beyond and prove these estimator's asymptotic normality when h goes to infinity, for any fixed value of k. The main ingredient is the asymptotic analysis of a functional equation on an appropriate characteristic function. Some numerical simulations illustrate the convergence to rely on Gaussian confidence intervals.