Moderate deviations for particle filtering

Consider the state space model (X _t, Y _t), where (X _t) is a Markov chain, and (Y _t) are the observations. In order to solve the so-called filtering problem, one has to compute ℒ(X _t|Y ₁,..., Y _t), the law of X _t given the observations (Y ₁,..., Y _t). The particle filtering method gives an approximation of the law ℒ(X _t|Y _t,...,Y _t) by an empirical measure 1/n∑ ₁ ⁿδ _xi,t In this paper we establish the moderate deviation principle for the empirical mean 1/n∑ ₁ ⁿψ(x _i,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.