Exponential stability in discrete-time filtering for non-ergodic signal.
A discrete-time filtering model is considered where the signal and the observation process are given as follows: \[ X_n=a(X_{n-1})+b(X_{n-1})\xi _n,\quad n\geq 1 \] and \[ Y_n=X_n+\sigma \nu _n,\quad n\geq 1 \] where \(X_0\) is an \(\mathbb R^d\)-valued random variable and the coefficients \(a:\mathbb R^d\to \mathbb R^d\) and \(b:\mathbb R^d \to \mathbb R^{d\times d}\) are measurable maps. It is shown that, under appropriate conditions on coefficients, the total variation distance between the optimal filter and an incorrectly initialized filter converges to zero almost surely exponentially fast as time approaches infinity if \(\sigma \) is sufficiently small.
scientific article; zbMATH DE number 1792521