Convergence Property of a Generic Particle Filter Algorithm

Particle filters are widely utilized in the optimal filtering problems. These methods approximate the posteriori distribution of the state (or the posteriori joint distribution of the extened state) by a population of weighted particles which evolve randomly according to the dynamic system model and the measurements. Despite many theoretical advance which have been reported in the last decade, the study of the convergence property of particle filters is still an open question. In this paper, the almost sure convergence of the generic particle filter (GPF) is discussed in a circuitous way. First, a modified-generic particle filter (M-GPF) is constructed. Different from the GPF, the M-GPF will determine whether it is necessary to rerun both the resampling step and the importance sampling (IS) step according to a conditional criterion after performing the IS step at each time. Then the almost sure convergence of the M-GPF will be concerned. Later, when the recursive time is finite and the interesting function is 4th power integrable with respect to the posteriori joint distribution of the extended state, the sufficient condition for the GPF estimation converges almost surely to the optimal estimation is discussed. Finally, a novel simulation experiment will be presented to illustrate the almost sure convergence of the GPF.