Moderate Deviations of the Random Riccati Equation

Abstract

We characterize the invariant filtering measures resulting from Kalman filtering with intermittent observations (\cite{Bruno}), where the observation arrival is modeled as a Bernoulli process. In \cite{Riccati-weakconv}, it was shown that there exists a $\overlineγ^{\{\scriptsize{sb}}}>0$ such that for every observation packet arrival probability $\overlineγ$, $\overlineγ>\overlineγ^{\{\scriptsize{sb}}}>0$, the sequence of random conditional error covariance matrices converges in distribution to a unique invariant distribution $\mathbbμ^{\overlineγ}$ (independent of the filter initialization.) In this paper, we prove that, for controllable and observable systems, $\overlineγ^{\{\scriptsize{sb}}}=0$ and that, as $\overlineγ\uparrow 1$, the family $\{\mathbbμ^{\overlineγ}\}_{\overlineγ>0}$ of invariant distributions satisfies a moderate deviations principle (MDP) with a good rate function $I$. The rate function $I$ is explicitly identified. In particular, our results show: