By making use of martingale representations, we derive the asymptotic normality of particle filters in hidden Markov models and a relatively simple formula for their asymptotic variances. Although repeated resamplings result in complicated dependence among the sample paths, the asymptotic variance formula and martingale representations lead to consistent estimates of the standard errors of the particle filter estimates of the hidden states.
"A general theory of particle filters in hidden Markov models and some applications." Ann. Statist. 41 (6) 2877 - 2904, December 2013. https://doi.org/10.1214/13-AOS1172