  In the first part of the paper, we consider a discrete-time stochastic
control system. We show that, under certain conditions, the set of random
occupational measures generated by the state-control trajectories of the system
as well as the set of their mathematical expectations converge (as the time
horizon tends to infinity) to a convex and compact (non-random) set, which is
shown to coincide with the set of stationary probabilities of the system. In
the second part, we apply the results obtained in the first part to deal with a
hybrid system that evolves in continuous time and is subjected to abrupt
changes of certain parameters. We show that the solutions of such a hybrid
system are approximated by the solutions of a differential inclusion, the
right-hand side of which is defined by the limit occupational measures set, the
existence and convexity of which is established in the first part of the paper.
