Sample-path average optimality for Markov control processes

The authors consider a Markov control process with Borel state and actions spaces, unbounded costs, and under the long-run sample-path average cost criterion. They prove that under very weak assumptions on the transition law and a moment assumption for the one-step cost, there exists a stationary policy with invariant probability distribution v, that is sample-path average cost optimal for v-almost all initial states. In addition, every expected average-cost optimal stationary policy is in fact (liminf) sample-path average-cost optimal and strongly expected average-cost optimal