  Multilevel Splitting is a Sequential Monte Carlo method to simulate
realisations of a rare event as well as to estimate its probability. This
article is concerned with the convergence and the fluctuation analysis of
Adaptive Multilevel Splitting techniques. In contrast to their fixed level
version, adaptive techniques estimate the sequence of levels on the fly and in
an optimal way, with only a low additional computational cost. However, very
few convergence results are available for this class of adaptive branching
models, mainly because the sequence of levels depends on the occupation
measures of the particle systems. This article proves the consistency of these
methods as well as a central limit theorem. In particular, we show that the
precision 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.
