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On non-ergodic versions of limit theorems. (English) Zbl 0707.60027
This paper is the very interesting result of combining a standard construction from ergodic theory - the ergodic decomposition of a strictly stationary but nonergodic process - with five known probabilistic limit theorems for stationary and ergodic martingale difference sequences. Two variants of each theorem are presented: one indicating the generalization via ergodic decomposition to nonergodic stationary martingale differences, the other the generalization (or counterexample) to the almost-sure version of the theorem for the “conditional stationary processes” induced by the ergodic decomposition.
The starting point is the author’s probabilistic reformulation of the ergodic decomposition. Using a regular conditional probability distribution given the invariant \(\sigma\)-field of a specified nonergodic stationary process on the product Borel probability space \(({\mathbb{R}}^{{\mathbb{Z}}},P)\), the author has elsewhere shown the existence of a family of regular conditional probabilities \(P_{\omega}\) with respect to almost each of which the shift-transformation T is measure- preserving and ergodic.
The limit theorems which the author considers are central and functional limit theorems for stationary square-integrable sequences of martingale differences, including some assertions that limits of variances or first absolute moments of normalized sums coincide with the corresponding moments of the asymptotic normal distribution.
Reviewer: E.Slud

MSC:
60F05 Central limit and other weak theorems
60F17 Functional limit theorems; invariance principles
60G10 Stationary stochastic processes
28D05 Measure-preserving transformations
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