  Ergodic properties and asymptotic stationarity are investigated in this paper
for the pseudo-covariance matrix (PCM) of a recursive state estimator which is
robust against parametric uncertainties and is based on plant output
measurements that may be randomly dropped. When the measurement dropping
process is described by a Markov chain and the modified plant is both
controllable and observable, it is proved that if the dropping probability is
less than 1, this PCM converges to a stationary distribution that is
independent of its initial values. A convergence rate is also provided. In
addition, it has also been made clear that when the initial value of the PCM is
set to the stabilizing solution of the algebraic Riccati equation related to
the robust state estimator without measurement dropping, this PCM converges to
an ergodic process. Based on these results, two approximations are derived for
the probability distribution function of the stationary PCM, as well as a bound
of approximation errors. A numerical example is provided to illustrate the
obtained theoretical results.
