On the convergence of sequences of stationary jump Markov processes

This paper presents two main results: first, a Liapunov type criterion for the existence of a stationary probability distribution for a jump Markov process; second, a Liapunov type criterion for existence and tightness of stationary probability distributions for a sequence of jump Markov processes. If the corresponding semigroups TN(t) converge, under suitable hypotheses on the limit semigroup, this last result yields the weak convergence of the sequence of stationary processes (TN(t), [pi]N) to the stationary limit one.