On non-ergodic versions of limit theorems.

*(English)*Zbl 0707.60027This 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.

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 |

##### Keywords:

central limit theorem for martingale differences; ergodic decomposition; invariance principle; invariant measure; law of iterated logarithm; strictly stationary sequence
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