Convergence of moments in a Markov-chain central limit theorem

Let (X-t)(i=0)(x) be a V-uniformly ergodic Markov chain on a general state space, and let pi be its stationary distribution. For g; X --> R, define W-k (g) := k(-1,2) Sigma(i-0)(k-1) (g(X-i) - pi(g)). It is shown that if \g\ less than or equal to V-I/n for a positive integer n, then E-x W-k (g)(n) converges to the n-th moment of a normal random variable with expectation 0 and variance gamma(g)(2) := pi(g(2)) - pi(g)(2) + Sigma(i-l)(x) (integralg(x)E(x)g(X-l)-pi(g)(2)). This extends the existing Markov-chain central limit theorems, according to which expectations of bounded functionals of W-k (g) converge. We also deride nonasymptotic bounds for the error in approximating the moments of W-k (g) by the normal moments. These yield easy bounds of all feasible poly normal orders, and exponential bounds as well under some circumstances, for the probabilities of large deviations by the empirical measure along the Markov chain path X-i.