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Central limit theorems for additive functionals of Markov chains. (English) Zbl 1044.60014

Summary: Central limit theorems and invariance principles are obtained for additive functionals of a stationary ergodic Markov chain, say \(S_n=g(X_1)+ \cdots+ g(X_n)\), where \(E[g(X_1)]=0\) and \(E[g (X_1)^2] <\infty\). The conditions imposed restrict the moments of \(g\) and the growth of the conditional means \(E(S_n\mid X_1)\). 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 \(g\).

MSC:

60F05 Central limit and other weak theorems
60J55 Local time and additive functionals
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60J05 Discrete-time Markov processes on general state spaces
60F17 Functional limit theorems; invariance principles
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