On non-ergodic versions of limit theorems

Dalibor Volný

Aplikace matematiky (1989)

  • Volume: 34, Issue: 5, page 351-363
  • ISSN: 0862-7940

Abstract

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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.

How to cite

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Volný, Dalibor. "On non-ergodic versions of limit theorems." Aplikace matematiky 34.5 (1989): 351-363. <http://eudml.org/doc/15589>.

@article{Volný1989,
abstract = {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.},
author = {Volný, Dalibor},
journal = {Aplikace matematiky},
keywords = {central limit theorem for martingale differences; ergodic decomposition; invariance principle; invariant measure; law of iterated logarithm; strictly stationary sequence; central limit theorem for martingale differences; ergodic decomposition; invariance principle; invariant measure; law of iterated logarithm; strictly stationary sequence},
language = {eng},
number = {5},
pages = {351-363},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On non-ergodic versions of limit theorems},
url = {http://eudml.org/doc/15589},
volume = {34},
year = {1989},
}

TY - JOUR
AU - Volný, Dalibor
TI - On non-ergodic versions of limit theorems
JO - Aplikace matematiky
PY - 1989
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 34
IS - 5
SP - 351
EP - 363
AB - 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.
LA - eng
KW - central limit theorem for martingale differences; ergodic decomposition; invariance principle; invariant measure; law of iterated logarithm; strictly stationary sequence; central limit theorem for martingale differences; ergodic decomposition; invariance principle; invariant measure; law of iterated logarithm; strictly stationary sequence
UR - http://eudml.org/doc/15589
ER -

References

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  10. C. C. Heyde, 10.1017/S0004972700023583, Bull. Austral. Math. Soc. 12 (1975), 1-8. (1975) Zbl0287.60035MR0372954DOI10.1017/S0004972700023583
  11. I. A. Ibragimov, A central limit theorem for a class of dependent random variables, Theory Probab. Appl. 8 (1963), 83-89. (1963) Zbl0123.36103MR0151997
  12. M. Loève, Probability Theory, Van Nostrand, New York, 1955. (1955) MR0203748
  13. J. C. Oxtoby, 10.1090/S0002-9904-1952-09580-X, Bull. Amer. Math. Soc. 58 (1952), 116-136. (1952) Zbl0046.11504MR0047262DOI10.1090/S0002-9904-1952-09580-X
  14. D. Volný, The central limit problem for strictly stationary sequences, Ph. D. Thesis, Mathematical Inst. Charles University, Praha, 1984. (1984) 
  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. (1986) MR0936028
  16. D. Volný, Martingale decompositions of stationary processes, Yokohama Math. J. 35 (1987), 113-121. (1987) MR0928378
  17. D. Volný, Counterexamples to the central limit problem for stationary dependent random variables, Yokohama Math. J. 36 (1988), 69-78. (1988) MR0978876
  18. D. Volný, On the invariance principle and functional law of iterated logarithm for non ergodic processes, Yokohama Math. J. 35 (1987), 137-141. (1987) MR0928380
  19. D. Volný, A non ergodic version of Gordin's CLT for integrable stationary processes, Comment. Math. Univ. Carolinae 28, 3 (1987), 419-425. (1987) MR0912569
  20. K. Winkelbauer, [unknown], personal communication. Zbl0584.94013

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