Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-28T03:29:37.178Z Has data issue: false hasContentIssue false
  • English
  • Français

Almost sure invariance principle for some maps of an interval

Published online by Cambridge University Press:  19 September 2008

Krystyna Ziemian
Krystyna Ziemian
Affiliation:
Instytut Matematyki, Uniwersytet Warszawski, PKiN IXp, 00-901 Warszawa, Poland
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove an almost sure invariance principle and a central limit theorem for the process , where f is a map of an interval with a non-positive Schwarzian derivative whose trajectories of critical points stay far from the critical points, and F is a measurable function with bounded p-variation (p ≥ 1).

The almost sure invariance principle implies the Log-log laws, integral tests and a distributional type of invariance principle for the process .

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 5 , Issue 4 , December 1985 , pp. 625 - 640
Copyright
Copyright © Cambridge University Press 1985

References

REFERENCES

[1]Bowen, R.. Bernoulli maps of an inverval. Israel J. Math. 28 (1977), 161–168.CrossRefGoogle Scholar
[2]Gordin, M. I.. The central limit theorem for stationary processes. Soviet Math. Dokl. 10 (1969), 1174–1176.Google Scholar
[3]Hofbauer, F., & Keller, G.. Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Zeit. 180 (1982), 119–140.CrossRefGoogle Scholar
[4]Keller, G.. Un theoreme de la Hmite central pour une class de transformations monotonnes par morceaux. C.R. Acad. Sc. Paris, Serie A, 291 (1980).Google Scholar
[5]Misiurewicz, M.. Absolutely continuous measures for certain maps of an inverval. Publ. Math. IHES 53 (1981), 17–51.CrossRefGoogle Scholar
[6]Phillipp, W. & Stout, W.. Almost sure invariance principles for partial sums of weakly dependent random variables. Mem. Amer. Math. Soc. 161 (1975).Google Scholar
[7]Szewc, B.. Perron-Frobenius operators in the spaces of smooth functions on an interval. Thesis.Google Scholar
[8]Szlenk, W.. Some dynamical properties of certain differentiable mappings of an inverval. Bol. de la Soc. Mex. 24, No. 2, (1979), 57–82.Google Scholar
[9]Wong, S.. A central limit theorem for piecewise monotonic mappings of the unit interval. Ann. Prob. (1979), 500–154.CrossRefGoogle Scholar