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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