Correlation bounds for discrete-time systems with banded dynamics

We consider the steady-state error covariance for a discrete-time system with banded dynamics. Such systems frequently arise from the spatial and temporal discretization of partial differential equations. In such systems, the magnitudes of the entries of the steady-state covariance matrix typically decrease as the distance from the diagonal increases. We obtain a bound on the entries of the covariance matrix beyond a given distance from the diagonal.