Unique Ergodicity of Non-Linear Filters via Reachability and Uniform Weak Continuity

TL;DR

This paper introduces a reachability-based method to prove unique ergodicity of non-linear filters in hidden Markov models with complex state and observation spaces, providing explicit conditions and convergence results.

Contribution

It develops a novel reachability approach for establishing unique ergodicity, complementing existing filter stability methods, applicable to models with Polish metric spaces.

Findings

Established a reachability condition for unique ergodicity.

Proved weak convergence of occupation measures under these conditions.

Provided explicit criteria applicable to complex state spaces.

Abstract

We present a reachability based approach to establish unique ergodicity of non-linear filter processes where state space of a hidden Markov model is a compact Polish metric space and the observation space is a Polish metric space. We also establish a weak convergence result on occupation measures under such a reachability condition. Our conditions, which are explicit, are complementary to those based on filter stability as demonstrated in examples.