The convergence of Cesaro averages for certain nonstationary Markov chains

If P is a stochastic matrix corresponding to a stationary, irreducible, positive persistent Markov chain of period d>1, the powers Pn will not converge as n --> [infinity]. However, the subsequences Pnd+k for k=0,1,...d-1, and hence Cesaro averages [Sigma]nk-1 Pk/n, will converge. In this paper we determine classes of nonstationary Markov chains for which the analogous subsequences and/or Cesaro averages converge and consider the rates of convergence. The results obtained are then applied to the analysis of expected average cost.