| Title:
|
On non-ergodic versions of limit theorems (English) |
| Author:
|
Volný, Dalibor |
| Language:
|
English |
| Journal:
|
Aplikace matematiky |
| ISSN:
|
0373-6725 |
| Volume:
|
34 |
| Issue:
|
5 |
| Year:
|
1989 |
| Pages:
|
351-363 |
| Summary lang:
|
English |
| Summary lang:
|
Russian |
| Summary lang:
|
Czech |
| . |
| Category:
|
math |
| . |
| Summary:
|
The author investigates non ergodic versions of several well known limit theorems for strictly stationary processes. In some cases, the assumptions which are given with respect to general invariant measure, guarantee the validity of the theorem with respect to ergodic components of the measure. In other cases, the limit theorem can fail for all ergodic components, while for the original invariant measure it holds. (English) |
| Keyword:
|
central limit theorem for martingale differences |
| Keyword:
|
ergodic decomposition |
| Keyword:
|
invariance principle |
| Keyword:
|
invariant measure |
| Keyword:
|
law of iterated logarithm |
| Keyword:
|
strictly stationary sequence |
| MSC:
|
28D05 |
| MSC:
|
60B10 |
| MSC:
|
60F05 |
| MSC:
|
60F17 |
| MSC:
|
60G10 |
| MSC:
|
60G40 |
| idZBL:
|
Zbl 0707.60027 |
| idMR:
|
MR1014076 |
| DOI:
|
10.21136/AM.1989.104363 |
| . |
| Date available:
|
2008-05-20T18:37:18Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/104363 |
| . |
| Reference:
|
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| Reference:
|
[2] P. Billingsley: The Lindenberg-Lévy theorem for martingales.Proc. Amer. Math. Soc. 12 (1961), 788-792. MR 0126871 |
| Reference:
|
[3] G. K. Eagleson: On Gordin's central limit theorem for stationary processes.J. Appl. Probab. 12 (1975), 176-179. Zbl 0306.60017, MR 0383501, 10.2307/3212425 |
| Reference:
|
[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. MR 0780729, 10.1016/0304-4149(85)90048-1 |
| Reference:
|
[5] M. I. Gordin: The central limit theorem for stationary processes.Soviet Math. Dokl. 10 (1969), 1174-1176. Zbl 0212.50005, MR 0251785 |
| Reference:
|
[6] M. I. Gordin: Abstracts of Communications, T.1: A-K.International conference on probability theory (Vilnius, 1973). |
| Reference:
|
[7] P. Hall C. C. Heyde: Martingale Limit Theory and its Applications.Academic Press, New York, 1980. MR 0624435 |
| Reference:
|
[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, MR 0517475, 10.2307/3213349 |
| Reference:
|
[9] C. C. Heyde: On the central limit theorem for stationary processes.Z. Wahrsch. Verw. Gebiete 30 (1974), 315-320. Zbl 0297.60014, MR 0372955, 10.1007/BF00532619 |
| Reference:
|
[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, MR 0372954, 10.1017/S0004972700023583 |
| Reference:
|
[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, MR 0151997 |
| Reference:
|
[12] M. Loève: Probability Theory.Van Nostrand, New York, 1955. MR 0203748 |
| Reference:
|
[13] J. C. Oxtoby: Ergodic sets.Bull. Amer. Math. Soc. 58 (1952), 116-136. Zbl 0046.11504, MR 0047262, 10.1090/S0002-9904-1952-09580-X |
| Reference:
|
[14] D. Volný: The central limit problem for strictly stationary sequences.Ph. D. Thesis, Mathematical Inst. Charles University, Praha, 1984. |
| Reference:
|
[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. MR 0936028 |
| Reference:
|
[16] D. Volný: Martingale decompositions of stationary processes.Yokohama Math. J. 35 (1987), 113-121. MR 0928378 |
| Reference:
|
[17] D. Volný: Counterexamples to the central limit problem for stationary dependent random variables.Yokohama Math. J. 36 (1988), 69-78. MR 0978876 |
| Reference:
|
[18] D. Volný: On the invariance principle and functional law of iterated logarithm for non ergodic processes.Yokohama Math. J. 35 (1987), 137-141. MR 0928380 |
| Reference:
|
[19] D. Volný: A non ergodic version of Gordin's CLT for integrable stationary processes.Comment. Math. Univ. Carolinae 28, 3 (1987), 419-425. MR 0912569 |
| Reference:
|
[20] K. Winkelbauer: .personal communication. Zbl 0584.94013 |
| . |
