Kalman Filtering with Intermittent Observations: Weak Convergence to a Stationary Distribution

TL;DR

This paper analyzes the long-term behavior of Kalman filtering with intermittent observations modeled as a Bernoulli process, showing weak convergence of the error covariance to a fractal-supported invariant distribution.

Contribution

It introduces a novel approach modeling the Riccati equation as an order-preserving, strongly sublinear random dynamical system and characterizes the invariant distribution's support.

Findings

Weak convergence to a unique invariant distribution occurs even below the critical observation rate.

The support of the invariant distribution exhibits fractal and self-similar properties.

Moments of the invariant distribution can be computed explicitly using the characterization of its support.

Abstract

The paper studies the asymptotic behavior of Random Algebraic Riccati Equations (RARE) arising in Kalman filtering when the arrival of the observations is described by a Bernoulli i.i.d. process. We model the RARE as an order-preserving, strongly sublinear random dynamical system (RDS). Under a sufficient condition, stochastic boundedness, and using a limit-set dichotomy result for order-preserving, strongly sublinear RDS, we establish the asymptotic properties of the RARE: the sequence of random prediction error covariance matrices converges weakly to a unique invariant distribution, whose support exhibits fractal behavior. In particular, this weak convergence holds under broad conditions and even when the observations arrival rate is below the critical probability for mean stability. We apply the weak-Feller property of the Markov process governing the RARE to characterize the support of the limiting invariant distribution as the topological closure of a countable set of points, which, in general, is not dense in the set of positive semi-definite matrices. We use the explicit characterization of the support of the invariant distribution and the almost sure ergodicity of the sample paths to easily compute the moments of the invariant distribution. A one dimensional example illustrates that the support is a fractured subset of the non-negative reals with self-similarity properties.