  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 $L^4$ 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
the boundedness of the importance weights, but only boundedness of second or
fourth order moments. We show that the boundedness of the second order moments
of the weights implies the convergence of the estimates bounded functions in
the mean square sense, and the $L^4$ convergence as well as the 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.
