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

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.

💡 Research Summary

The paper investigates the long‑term behavior of the error‑covariance recursion that arises in a Kalman filter when measurements arrive intermittently according to an i.i.d. Bernoulli process. In this setting the covariance update can be written as a random algebraic Riccati equation (RARE)
(P_{k+1}=f_{\gamma_k}(P_k)),
where (\gamma_k\in{0,1}) indicates whether a measurement is received at time (k). The two deterministic maps (f_0) (no measurement) and (f_1) (measurement) are both order‑preserving and strongly sublinear on the cone of positive semidefinite matrices. By interpreting the sequence ({P_k}) as a random dynamical system (RDS) with these properties, the authors bring to bear a powerful limit‑set dichotomy theorem for order‑preserving, strongly sublinear RDS.

The dichotomy states that, under a mild “stochastic boundedness” condition, the RDS either diverges to infinity on every trajectory or converges weakly to a unique invariant probability measure (\mu). The paper shows that stochastic boundedness holds whenever the underlying plant ((A,Q)) is stable and the measurement arrival probability (p>0). Consequently, the divergent case is ruled out and the covariance sequence converges in distribution to a single invariant law (\mu), regardless of the initial covariance.

The invariant law (\mu) is shown to be a fixed point of a weak‑Feller Markov kernel associated with the RARE. Using the weak‑Feller property, the authors characterize the support of (\mu) explicitly as the topological closure of the countable set
({f_{i_1}\circ\cdots\circ f_{i_m}(P^\star)\mid m\in\mathbb{N},; i_j\in{0,1}}),
where (P^\star) is the steady‑state covariance when measurements are always available. This set is typically fractal: it is countable before closure, and after closure it exhibits self‑similar gaps and is not dense in the whole cone of positive semidefinite matrices. Hence the limiting distribution lives on a “fractured” subset of the state‑covariance space.

Because the Markov process governing ({P_k}) is ergodic under (\mu), time averages along a single sample path converge almost surely to expectations with respect to (\mu). This almost‑sure ergodicity makes it possible to compute moments of the invariant distribution simply by long‑run simulation of one trajectory, without the need for ensemble averaging.

A scalar example (with (A=\sqrt{2}, Q=1, C=1, R=1) and measurement arrival probability (p=0.4)) illustrates the theory. Although (p) is below the critical probability for mean‑square stability, the covariance still converges weakly to a distribution whose support is a Cantor‑like subset of (\mathbb{R}_+). The support displays self‑similarity with scaling factor (\sqrt{2}), confirming the fractal nature predicted by the analysis.

The main contributions of the work are: (1) a rigorous proof that intermittent‑observation Kalman filters possess a unique invariant distribution under very general conditions, even when the average arrival rate is insufficient for mean‑square stability; (2) a detailed description of the invariant distribution’s support as a fractal closure of a countable orbit set; (3) the exploitation of weak‑Feller and ergodic properties to enable straightforward moment computation; and (4) a demonstration that the support is typically a proper, non‑dense subset of the positive semidefinite cone, contrary to intuition that the filter might explore the whole space.

These results broaden the understanding of stability for networked control and estimation systems where packet loss, scheduling, or energy constraints cause measurements to be sporadic. They suggest that design criteria based on stochastic boundedness rather than mean‑square stability may be sufficient in many practical scenarios. Future work could extend the analysis to Markov‑modulated observation processes, nonlinear dynamics, or non‑Gaussian noise, where the order‑preserving and sublinear structure may still be leveraged to obtain analogous invariant‑distribution results.