  This paper investigates the stability of Kalman filtering over
Gilbert-Elliott channels where random packet drop follows a time-homogeneous
two-state Markov chain whose state transition is determined by a pair of
failure and recovery rates. First of all, we establish a relaxed condition
guaranteeing peak-covariance stability described by an inequality in terms of
the spectral radius of the system matrix and transition probabilities of the
Markov chain. We further show that that condition can be interpreted using a
linear matrix inequality feasibility problem. Next, we prove that the
peak-covariance stability implies mean-square stability, if the system matrix
has no defective eigenvalues on the unit circle. This connection between the
two stability notions holds for any random packet drop process. We prove that
there exists a critical curve in the failure-recovery rate plane, below which
the Kalman filter is mean-square stable and no longer mean-square stable above,
via a coupling method in stochastic processes. Finally, a lower bound for this
critical failure rate is obtained making use of the relationship we establish
between the two stability criteria, based on an approximate relaxation of the
system matrix.
