Limiting distributions of functionals of Markov chains

Let \s{Xn, n [greater-or-equal, slanted] 0\s} and \s{Yn, n [greater-or-equal, slanted] 0\s} be two stochastic processes such that Yn depends on Xn in a stationary manner, i.e. P(Yn [epsilon] A\vbXn) does not depend on n. Sufficient conditions are derived for Yn to have a limiting distribution. If Xn is a Markov chain with stationary transition probabilities and Yn = f(Xn,..., Xn+k) then Yn depends on Xn is a stationary way. Two situations are considered: (i) \s{Xn, n [greater-or-equal, slanted] 0\s} has a limiting distribution (ii) \s{Xn, n [greater-or-equal, slanted] 0\s} does not have a limiting distribution and exits every finite set with probability 1. Several examples are considered including that of a non-homogeneous Poisson process with periodic rate function where we obtain the limiting distribution of the interevent times.