A particle system is a collection where are “particles” and are their “weights”. The system targets a distribution on if for any measurable with ,
A sequential Monte Carlo algorithm (a particle filter) produces recursively (using mutation-correction-resampling scheme) a sequence of particle systems which target a sequence of distributions on . In the Bayes estimation problems is the parameter space and is an a posteriori distribution of the parameter given the sample of size . In the state-space filtering or smoothing is the space of states trajectories and is the conditional distribution of the trajectory given the data.
The author obtains conditions for the central limit theorem of the form where is described using recursive formulae. These conditions hold for many of sequential Monte Carlo algorithms including the resample-move algorithm and the residual resampling scheme. Asymptotics of as are investigated for Bayesian problems.