  It is known that state-dependent, multi-step Lyapunov bounds lead to greatly
simplified verification theorems for stability for large classes of Markov
chain models. This is one component of the "fluid model" approach to stability
of stochastic networks. In this paper we extend the general theory to
randomized multi-step Lyapunov theory to obtain criteria for stability and
steady-state performance bounds, such as finite moments.
  These results are applied to a remote stabilization problem, in which a
controller receives measurements from an erasure channel with limited capacity.
Based on the general results in the paper it is shown that stability of the
closed loop system is assured provided that the channel capacity is greater
than the logarithm of the unstable eigenvalue, plus an additional correction
term. The existence of a finite second moment in steady-state is established
under additional conditions.
