Uniform Time Average Consistency of Monte Carlo Particle Filters

TL;DR

This paper proves that bootstrap Monte Carlo particle filters can uniformly approximate the optimal nonlinear filter over time under certain ergodicity and regularity conditions, ensuring reliable long-term filtering performance.

Contribution

It establishes uniform time average consistency of particle filters for ergodic signals, including noncompact cases with geometric ergodicity and regular observations.

Findings

Uniform approximation holds over time horizon for ergodic signals.

Consistency achieved without extra assumptions in compact state spaces.

Results extend to noncompact spaces with geometric ergodicity and regular observations.

Abstract

We prove that bootstrap type Monte Carlo particle filters approximate the optimal nonlinear filter in a time average sense uniformly with respect to the time horizon when the signal is ergodic and the particle system satisfies a tightness property. The latter is satisfied without further assumptions when the signal state space is compact, as well as in the noncompact setting when the signal is geometrically ergodic and the observations satisfy additional regularity assumptions.