Central limit theorems for additive functionals of Markov chains

Abstract

Central limit theorems and invariance principles are obtained for additive functionals of a stationary ergodic Markov chain, say Sn = g(X1) + ⋯ + g(Xn), where E[g(X1)] = 0 and E[g(X1)2] < ∞. The conditions imposed restrict the moments of g and the growth of the conditional means E(Sn | X1). No other restrictions on the dependence structure of the chain are required. When specialized to shift processes, the conditions are implied by simple integral tests involving g.