Stability properties of some particle filters

TL;DR

This paper proves that under certain conditions, particle filters maintain bounded variance over time, ensuring reliable long-term performance in estimating hidden Markov models.

Contribution

It establishes uniform and linear bounds on variances of particle filter estimates under regularity conditions, extending stability analysis to noncompact state spaces.

Findings

Asymptotic variance is uniformly bounded in time.

Nonasymptotic relative variance grows linearly with time.

Results apply to some hidden Markov models on noncompact spaces.

Abstract

Under multiplicative drift and other regularity conditions, it is established that the asymptotic variance associated with a particle filter approximation of the prediction filter is bounded uniformly in time, and the nonasymptotic, relative variance associated with a particle approximation of the normalizing constant is bounded linearly in time. The conditions are demonstrated to hold for some hidden Markov models on noncompact state spaces. The particle stability results are obtained by proving $v$-norm multiplicative stability and exponential moment results for the underlying Feynman-Kac formulas.