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Limit theorems for functionals of ergodic Markov chains with general state space. (English) Zbl 0952.60014

Mem. Am. Math. Soc. 664, 203 p. (1999).
Author’s abstract: Let \(\{X_n\}_{n\geq 1}\) be an ergodic Markov chain with general state space \(E\) and let \(\xi\) be a measurable map from \(E\) to \({\mathbb R}\) or to some Banach space \(B\) and set \(S_n=\sum_{k=0}^{n-1}\xi(X_k)\), \(n=1,2,\dots\;\). The central limit theorem, the law of the iterated logarithm, and the moderate deviation principle associated with \(\{S_n\}\) are established in a reasonable and natural way. Our arguments mainly depend on the split chain and regeneration methods which are systematically developed. All conditions appear to be the best possible (some even necessary) for our theorems and what is important, they are given in terms of the original chain.

MSC:

60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60F05 Central limit and other weak theorems
60F10 Large deviations
60F15 Strong limit theorems
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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