Moderate deviations for particle filtering

Abstract

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|Y₁,…,Y_t), the law of X_t given the observations (Y₁,…,Y_t). The particle filtering method gives an approximation of the law ℒ(X_t|Y₁,…,Y_t) by an empirical measure $\frac{1}{n}$∑₁ⁿδ_xi,t. In this paper we establish the moderate deviation principle for the empirical mean $\frac{1}{n}$∑₁ⁿψ(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.