  The forgetting of the initial distribution for discrete Hidden Markov Models
(HMM) is addressed: a new set of conditions is proposed, to establish the
forgetting property of the filter, at a polynomial and geometric rate. 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 different HMM of interest: the
dynamic tobit model, the non-linear state space model and the stochastic
volatility model.
