Unbiasedness of some generalized adaptive multilevel splitting algorithms

We introduce a generalization of the Adaptive Multilevel Splitting algorithm in the discrete time dynamic setting, namely when it is applied to sample rare events associated with paths ofMarkov chains.We build an estimator of the rare event probability (and of any nonnormalized quantity associated with this event) which is unbiased, whatever the choice of the importance function and the number of replicas. This has practical consequences on the use of this algorithm, which are illustrated through various numerical experiments.