Denumerable controlled Markov chains with average reward criterion: sample path optimality

There are numerous applications, in many different fields, of denumerable controlled Markov chain (CMC) models with an infinite planning horizon; see Bertsekas (1987), Ephremides and Verdu (1989), Ross (1983), Stidham and Weber (1993), and Tijms (1986). The authors consider the stochastic control problem of maximizing average rewards in the long-run, for denumerable CMCs. Departing from the most common position which uses expected values of rewards, the authors focus on a sample path analysis of the stream of states and actions. Under a Lyapunov function condition, the authors show that stationary policies obtained from the average reward optimality equation are not only expected average reward optimal, but indeed sample path average reward optimal