Moderate Deviations of a Random Riccati Equation

Abstract

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 γ. Our prior work showed that, for γ >; 0 , the sequence of random conditional error covariance matrices converges weakly to a unique invariant distribution μγ. This paper shows that, as γ approaches one, the family {μγ}γ >; 0 satisfies a moderate deviations principle with good rate function I (·): (1) as γ ↑ 1 , the family {μγ} converges weakly to the Dirac measure δP* 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 μγ decays to zero as a power law of (1-γ) as γ↑ 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 μγ.