"Computing smoothing distributions, the distributions of one or more states conditional on past, present, and future observations is a recurring problem when operating on general hidden Markov models. The aim of this paper is to provide a foundation of particle-based approximation of such distributions and to analyze, in a common unifying framework, different schemes producing such approximations. In this setting, general convergence results, including exponential deviation inequalities and central limit theorems, are established. In particular, time uniform bounds on the marginal smoothing error are obtained under appropriate mixing conditions on the transition kernel of the latent chain. In addition, we propose an algorithm approximating the joint smoothing distribution at a cost that grows only linearly with the number of particles."

"Let $(Y_k)$ be a stationary sequence on a probability space taking values in a standard Borel space. Consider the associated maximum likelihood estimator with respect to a parametrized family of Hidden Markov models such that the law of the observations $(Y_k)$ is not assumed to be described by any of the Hidden Markov models of this family. In this paper we investigate the consistency of this estimator in such mispecified models under mild assumptions."