Recursibility and optimal linear estimation and filtering

It is well known that the Kalman filter is the recursive linear minimum mean-square error (LMMSE) filter for a linear system with some assumptions on auto- and cross-correlations of process and measurement noise and initial state. It is little known, however, that for many linear systems the LMMSE filter does not have a recursive form. This paper introduces the concept of recursibility and presents related results for optimal linear estimation and filtering for arbitrary auto- and cross-correlations of the noise and state without the Kalman filter assumptions. Specifically, we present necessary and sufficient conditions for the recursibility of LMMSE estimation and filtering; more important, we present recursive LMMSE estimators and filters that are not necessarily equivalent to the batch LMMSE estimators and filters, but are optimal within the recursive class.