Hostname: page-component-74656f9884-54qdv Total loading time: 0 Render date: 2026-11-09T20:13:19.836Z Has data issue: false hasContentIssue false
  • English
  • Français

Variations around Eagleson’s theorem on mixing limit theorems for dynamical systems

Part of: Ergodic theory

Published online by Cambridge University Press:  26 June 2019

SÉBASTIEN GOUËZEL
SÉBASTIEN GOUËZEL*
Affiliation:
Laboratoire Jean Leray, CNRS UMR 6629, Université de Nantes, 2 rue de la Houssinière, 44322Nantes, France email sebastien.gouezel@univ-nantes.fr
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.

Eagleson’s theorem asserts that, given a probability-preserving map, if renormalized Birkhoff sums of a function converge in distribution, then they also converge with respect to any probability measure which is absolutely continuous with respect to the invariant one. We prove a version of this result for almost sure limit theorems, extending results of Korepanov. We also prove a version of this result, in mixing systems, when one imposes a conditioning both at time 0 and at time $n$.

Keywords

Eagleson’s theoremalmost sure invariance principlelimit theoremsmixing of all orders

MSC classification

Primary: 37A25: Ergodicity, mixing, rates of mixing 37A40: Nonsingular (and infinite-measure preserving) transformations 37A50: Relations with probability theory and stochastic processes

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 12 , December 2020 , pp. 3368 - 3374
Check for updates
Copyright
© Cambridge University Press, 2019

References

Aaronson, J.. The asymptotic distributional behaviour of transformations preserving infinite measures. J. Anal. Math. 39 (1981), 203–234.Google Scholar
Eagleson, G. K.. Some simple conditions for limit theorems to be mixing. Teor. Veroyatn. Primen. 21 (1976), 653–660.Google Scholar
Korepanov, A.. Equidistribution for nonuniformly expanding dynamical systems, and application to the almost sure invariance principle. Comm. Math. Phys. 359(3) (2018), 1123–1138.Google Scholar
Melbourne, I. and Nicol, M.. Almost sure invariance principle for nonuniformly hyperbolic systems. Comm. Math. Phys. 260 (2005), 131–146.Google Scholar
Zweimüller, R.. Mixing limit theorems for ergodic transformations. J. Theoret. Probab. 20 (2007), 1059–1071.Google Scholar