  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.
