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Approximating martingales and the central limit theorem for strictly stationary processes. (English) Zbl 0765.60025

Consider a space \((\Omega,{\mathcal A},\mu)\) and a measure preserving transformation \(T:\Omega\to\Omega\). Let \(f\in L_ 2(\mu)\) and define \(s_ n(f)={1\over\sqrt n}\sum^ n_{j=1}f\circ T^ j\). The author studies assumptions of various central limit theorems (like those of Gordin and Dürr-Goldstein) which ensure that \(s_ n(f)\) weakly converges to a normal law. The theorems can be characterized by the fact that they guarantee \(\lim_{n\to\infty}\| s_ n(f-m)\|_ 2=0\) for any martingale difference \(m\circ T^ i\). Then the author discusses conditions under which the results hold in almost all ergodic components simultaneously. An application to the asymptotic theory of stationary linear processes with random coefficient is given.
Reviewer: J.Anděl (Praha)

MSC:

60G10 Stationary stochastic processes
60F05 Central limit and other weak theorems
60G42 Martingales with discrete parameter
60F17 Functional limit theorems; invariance principles
28D05 Measure-preserving transformations
60F15 Strong limit theorems
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