  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.
