2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/69791Consider 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_1,...,Y_t) by an empirical measure \frac{1}{n}\sum_1^nδ_{x_{i,t}}. In this paper we establish the moderate deviation principle for the empirical mean \frac{1}{n}\sum_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.Published at http://dx.doi.org/10.1214/105051604000000657 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)Probability60F10, 60G35, 93E11 (Primary)text