A lognormal central limit theorem for particle approximations of normalizing constants

Feynman-Kac models arise in a large variety of scientific disciplines including physics, chemistry and signal processing. Their mean field particle interpretations, termed commonly Sequential Monte Carlo or Particle Filters, have found numerous applications as they allow to sample approximately from sequences of complex probability distributions and estimate their associated normalizing constants. It is well-known that, under regularity assumptions, the relative variance of these normalizing constant estimates increases linearly with the time horizon n so that practitioners usually scale the number of particles N linearly w.r.t n to obtain estimates whose relative variance remains uniformly bounded w.r.t n. We establish here that, under this standard linear scaling strategy, the fluctuations of the normalizing constant estimates are lognormal as n, hence N, goes to infinity. For particle absorption models in a time-homogeneous environment and hidden Markov models in an ergodic random environment, we also provide more explicit descriptions of the limiting bias and variance.