Limiting distributions of functionals of Markov chains

Abstract

Let \s{X_n, n □ 0\s} and \s{Y_n, n □ 0\s} be two stochastic processes such that Y_n depends on X_n in a stationary manner, i.e. P(Y_n □ A\vbX_n) does not depend on n. Sufficient conditions are derived for Y_n to have a limiting distribution. If X_n is a Markov chain with stationary transition probabilities and Y_n = f(X_n,..., X_n+k) then Y_n depends on X_n is a stationary way. Two situations are considered: (i) \s{X_n, n □ 0\s} has a limiting distribution (ii) \s{X_n, n □ 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.