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In this paper, we derive moment conditions for particle filter importanceweights, which ensure that the particle filter estimates of the expectations ofbounded Borel functions converge in mean square and $L^4$ sense, and that theempirical measure of the particle filter converges weakly to the true filteringmeasure. The result extends the previously derived conditions by not requiringthe boundedness of the importance weights, but only boundedness of second orfourth order moments. We show that the boundedness of the second order momentsof the weights implies the convergence of the estimates bounded functions inthe mean square sense, and the $L^4$ convergence as well as the empiricalmeasure convergence are assured by the boundedness of the fourth order momentsof the weights. We also present an example class of models and importancedistributions where the moment conditions hold, but the boundedness does not.The unboundedness in these models is caused by point-singularities in theweights which still leave the weight moments bounded. We show by usingsimulated data that the particle filter for this kind of model also performswell in practice.

Moment Conditions for Convergence of Particle Filters with Unbounded Importance Weights