Fluctuation Analysis of Adaptive Multilevel Splitting

The purpose of this paper is to present a law of large numbers and a central limit theorem for Adaptive Multilevel Splitting algorithms. In rare event estimation, Multilevel Splitting is a sequential Monte Carlo method to estimate the probability of a rare event as well as to simulate realisations of this event. Contrarily to the fixed-levels version of Multilevel Splitting, where the successive levels are predefined, the adaptive version of this algorithm estimates the sequence of levels on the fly and in an optimal way at the price of a low additional computational cost. However, if a lot of results are available for the fixed-levels version thanks to a connection with the general framework of Feynman-Kac formulae, this is unfortunately not the case for the adaptive version. Hence, the aim of the present article is to go one step forward in the understanding of this practical and efficient method, at least from the law of large numbers and central limit viewpoints. In particular, we show that the asymptotic variance of the adaptive version is the same as the one of the fixed-levels version where the levels would have been placed in an optimal manner.