Non-asymptotic deviation inequalities for smoothed additive functionals in nonlinear state-space models

Abstract

The approximation of fixed-interval smoothing distributions is a key issue in inference for general state-space hidden Markov models (HMM). This contribution establishes non-asymptotic bounds for the Forward Filtering Backward Smoothing (FFBS) and the Forward Filtering Backward Simulation (FFBSi) estimators of fixed-interval smoothing functionals. We show that the rate of convergence of the $\mathrm{L}_{q}$-mean errors of both methods depends on the number of observations $T$ and the number of particles $N$ only through the ratio $T/N$ for additive functionals. In the case of the FFBS, this improves recent results providing bounds depending on $T/\sqrt{N}$.