Exponential stability in discrete-time filtering for non-ergodic signals

In this paper we prove exponential asymptotic stability for discrete-time filters for signals arising as solutions of d-dimensional stochastic difference equations. The observation process is the signal corrupted by an additive white noise of sufficiently small variance. The model for the signal admits non-ergodic processes. We show that almost surely, the total variation distance between the optimal filter and an incorrectly initialized filter converges to 0 exponentially fast as time approaches [infinity].