The Annals of Probability

Central limit theorems for additive functionals of Markov chains

Michael Maxwell and Michael Woodroofe

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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 $E(S_n|X_1)$. 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 $g$.

Article information

Ann. Probab., Volume 28, Number 2 (2000), 713-724.

First available in Project Euclid: 18 April 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems

Asymptotic normality ergodic theorem functional central limit theorem Hilbert space martingale maximal inequality one-sided shifts Poisson’s equation


Maxwell, Michael; Woodroofe, Michael. Central limit theorems for additive functionals of Markov chains. Ann. Probab. 28 (2000), no. 2, 713--724. doi:10.1214/aop/1019160258.

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