Moderate Deviations of a Random Riccati Equation

The paper characterizes the invariant filtering measures resulting from Kalman filtering with intermittent observations in which the observation arrival is modeled as a Bernoulli process with packet arrival probability γ̅. Our prior work showed that, for γ̅ >; 0 , the sequence of random conditional error covariance matrices converges weakly to a unique invariant distribution μ^γ̅. This paper shows that, as γ̅ approaches one, the family {μ^γ̅}_{γ̅ >; 0} satisfies a moderate deviations principle with good rate function I (·): (1) as γ̅ ↑ 1 , the family {μ^γ̅} converges weakly to the Dirac measure δ_P* concentrated on the fixed point of the associated discrete time Riccati operator; (2) the probability of a rare event (an event bounded away from P*) under μ^γ̅ decays to zero as a power law of (1-γ̅) as γ̅↑ 1; and, (3) the best power law decay exponent is obtained by solving a deterministic variational problem involving the rate function I (·). For specific scenarios, the paper develops computationally tractable methods that lead to efficient estimates of rare event probabilities under μ^γ̅.