Weight moment conditions for L4 convergence of particle filters for unbounded test functions

Particle filters are important approximation methods for solving probabilistic optimal filtering problems on nonlinear non-Gaussian dynamical systems. In this paper, we derive novel moment conditions for importance weights of sequential Monte Carlo based particle filters, which ensure the L convergence of particle filter approximations of unbounded test functions. This paper extends the particle filter convergence results of Hu & Schön & Ljung (2008) and Mbalawata & Särkkä (2014) by allowing for a general class of potentially unbounded importance weights and hence more general importance distributions. The result shows that provided that the seventh order moment is finite, then a particle filter for unbounded test functions with unbounded importance weights are ensured to converge.