Forgetting of the initial distribution for non-ergodic Hidden Markov Chains

TL;DR

This paper investigates how the influence of initial distribution diminishes over time in non-ergodic Hidden Markov Models, introducing new conditions that extend existing theoretical results on filter forgetting.

Contribution

It proposes a novel set of conditions that significantly extend previous results on the forgetting property in non-ergodic HMMs, covering both pathwise and expectation convergence.

Findings

Extended the theoretical understanding of filter forgetting in non-ergodic HMMs.

Provided new conditions ensuring convergence of the filter regardless of initial distribution.

Illustrated results with generic models demonstrating the applicability of the conditions.

Abstract

In this paper, the forgetting of the initial distribution for a non-ergodic Hidden Markov Models (HMM) is studied. A new set of conditions is proposed to establish the forgetting property of the filter, which significantly extends all the existing results. Both a pathwise-type convergence of the total variation distance of the filter started from two different initial distributions, and a convergence in expectation are considered. The results are illustrated using generic models of non-ergodic HMM and extend all the results known so far.