Convergence in distribution for filtering processes associated to Hidden Markov Models with densities

TL;DR

This paper proves that under certain ergodic and coupling conditions, the distribution of filtering processes in hidden Markov models converges and becomes independent of initial states, with stronger results under uniform ergodicity.

Contribution

It establishes convergence in distribution for filtering processes in hidden Markov models with densities, extending previous results to more general state and observation spaces.

Findings

Filtering process distribution becomes independent of initial distribution.

Convergence in distribution is achieved under strong ergodicity.

Uniform ergodicity ensures convergence in distribution.

Abstract

Consider a filtering process associated to a hidden Markov model with densities for which both the state space and the observation space are complete, separable, metric spaces. If the underlying, hidden Markov chain is strongly ergodic and the filtering process fulfills a certain coupling condition we prove that, in the limit, the distribution of the filtering process is independent of the initial distribution of the hidden Markov chain. If furthermore the hidden Markov chain is uniformly ergodic, then we prove that the filtering process converges in distribution.