On the law of large numbers for (geometrically) ergodic Markov chains

For use in asymptotic analysis of nonlinear time series models, we show that with (X-t, t >= 0) a (geometrically) ergodic Markov chain, the general version of the strong law of large numbers applies. That is, the average (1/T) Sigma(T-1)(t=O) phi(X-t,Xt+1,...) converges almost surely to the expectation of phi(XtXt+1,....) irrespective of the choice of initial distribution of, or value of, X-0. In the existing literature, the less general form (1/T) Sigma(T-1)(t=0) phi(X-t) has been studied.