Moderate deviations for particle filtering

Consider the state space model (Xt,Yt), where (Xt) is a Markov chain, and (Yt) are the observations. In order to solve the so-called filtering problem, one has to compute .(Xt|Y1,.,Yt), the law of Xt given the observations (Y1,.,Yt). The particle filtering method gives an approximation of the law .(Xt|Y1,.,Yt) by an empirical measure 1n.1n.xi,t. In this paper we establish the moderate deviation principle for the empirical mean 1n.1n.(xi,t) (centered and properly rescaled) when the number of particles grows to infinity, enhancing the central limit theorem. Several extensions and examples are also studied.