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 ~Xt , t 0! a ~geometrically! ergodic Markov chain, the general version of the strong law of large numbers applies+ That is, the average ~10T !(t 0 T 1 f~Xt , Xt 1, + + +! converges almost surely to the expectation of f~Xt , Xt 1, + + +! irrespective of the choice of initial distribution of, or value of, X0+ In the existing literature, the less general form ~10T !(t 0 T 1 f~Xt ! has been studied+