A note about the filtering problem in discrete time making use of weak convergence of probability measures.

In this paper we consider a partially observable stochastic process (X(n), Y-n), where X(n) is a Markov process and Y-n depends only on the current value of X(n). It is known that, under suitable regularity assumptions, if (X(n), Y-n) admits a finite-dimensional filter, then the prediction, observation, and filtering densities are all of exponential class. We prove that, under some additional assumptions, the existence of a finite-dimensional filter implies that the Markov transition kernel of X(n) is itself of exponential class. To do this we apply a simple property of weak convergence of probability measures.