Moment conditions for convergence of particle filters with unbounded importance weights

• We analyze the theoretical convergence of particle filter algorithm.
• We derive a novel mean square error (L2) convergence theorem for particle filters.
• The L2-convergence results is generalized to L4- and empirical measure convergence.
• We present an example where the moment conditions hold but the weights are unbounded.

In this paper, we derive moment conditions for particle filter importance weights, which ensure that the particle filter estimates of the expectations of bounded Borel functions converge in mean square and L4 sense, and that the empirical measure of the particle filter converges weakly to the true filtering measure. The result extends the previously derived conditions by not requiring boundedness of the importance weights, but only boundedness of the second or fourth order moments. We show that the boundedness of the second order moments of the weights implies the convergence of the particle filter for bounded test functions in the mean square sense. The L4 convergence and empirical measure convergence are assured by the boundedness of the fourth order moments of the weights. We also present an example class of models and importance distributions where the moment conditions hold, but the boundedness does not. The unboundedness in these models is caused by point-singularities in the weights which still leave the weight moments bounded. We show by using simulated data that the particle filter for this kind of model also performs well in practice.