# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] P. Billingsley: Ergodic Theory and Information. Wiley, New York) · Zbl 0184.43301 [2] P. Billingsley: The Lindenberg-Lévy theorem for martingales. Proc. Amer. Math. Soc. 12 (1961), 788-792. · Zbl 0129.10701 [3] G. K. Eagleson: On Gordin’s central limit theorem for stationary processes. J. Appl. Probab. 12 (1975), 176-179. · Zbl 0306.60017 [4] C. G. Esseen S. Janson: On moment conditions for normed sums of independent variables and martingale differences. Stoch. Proc. and their Appl. 19 (1985). 173-182. · Zbl 0554.60050 [5] M. I. Gordin: The central limit theorem for stationary processes. Soviet Math. Dokl. 10 (1969), 1174-1176. · Zbl 0212.50005 [6] M. I. Gordin: Abstracts of Communications, T.1: A-K. International conference on probability theory [7] P. Hall C. C. Heyde: Martingale Limit Theory and its Applications. Academic Press, New York, 1980. · Zbl 0462.60045 [8] C. C. Heyde: On central limit and iterated logarithm supplements to the martingale convergence theorem. J. Appl. Probab. 14 (1977), 758-775. · Zbl 0385.60033 [9] C. C. Heyde: On the central limit theorem for stationary processes. Z. Wahrsch. Verw. Gebiete 30 (1974), 315-320. · Zbl 0297.60014 [10] C. C. Heyde: On the central limit theorem and iterated logarithm law for stationary processes. Bull. Austral. Math. Soc. 12 (1975), 1-8. · Zbl 0287.60035 [11] I. A. Ibragimov: A central limit theorem for a class of dependent random variables. Theory Probab. Appl. 8 (1963), 83-89. · Zbl 0123.36103 [12] M. Loève: Probability Theory. Van Nostrand, New York, 1955. · Zbl 0066.10903 [13] J. C. Oxtoby: Ergodic sets. Bull. Amer. Math. Soc. 58 (1952), 116-136. · Zbl 0046.11504 [14] D. Volný: The central limit problem for strictly stationary sequences. Ph. D. Thesis, Mathematical Inst. Charles University, Praha, 1984. · Zbl 0568.60023 [15] D. Volný: Approximation of stationary processes and the central limit problem. LN in Mathematics 1299 (Proceedings of the Japan- USSR Symposium on Probability Theory, Kyoto 1986) 532-540. [16] D. Volný: Martingale decompositions of stationary processes. Yokohama Math. J. 35 (1987), 113-121. · Zbl 0636.60034 [17] D. Volný: Counterexamples to the central limit problem for stationary dependent random variables. Yokohama Math. J. 36 (1988), 69-78. [18] D. Volný: On the invariance principle and functional law of iterated logarithm for non ergodic processes. Yokohama Math. J. 35 (1987), 137-141. · Zbl 0638.60048 [19] D. Volný: A non ergodic version of Gordin’s CLT for integrable stationary processes. Comment. Math. Univ. Carolinae 28, 3 (1987), 419-425. · Zbl 0629.60030 [20] K. Winkelbauer: personal communication. · Zbl 0584.94013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.