Analysis of adaptive multilevel splitting algorithms in an idealized case

The Adaptive Multilevel Splitting algorithm [F. Cérou and A. Guyader, Stoch. Anal. Appl. 25 (2007) 417-443] is a very powerful and versatile method to estimate rare events probabilities. It is an iterative procedure on an interacting particle system, where at each step, the k less well-adapted particles among n are killed while k new better adapted particles are resampled according to a conditional law. We analyze the algorithm in the idealized setting of an exact resampling and prove that the estimator of the rare event probability is unbiased whatever k. We also obtain a precise asymptotic expansion for the variance of the estimator and the cost of the algorithm in the large n limit, for a fixed k.