On Convergence of Recursive Monte Carlo Filters in Non-Compact State Spaces

We consider the particle filter approximation of the optimal filter in non-compact state space models. A time-uniform convergence result is built on top of a filter stability argument developed by Douc, Moulines, and Ritov (2009), under the assumption of a heavy-tailed state process and an informative observation model. We show that an existing set of sufficient conditions for filter stability is also sufficient, with minor modifications, for particle filter convergence. The rate of convergence is also given and depends on both the sample size and the tail behavior of the transition kernel.