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

Published online by Cambridge University Press:  26 June 2019

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
Corresponding

Abstract

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 $n$.

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Original Article
Copyright
© Cambridge University Press, 2019

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References

Aaronson, J.. The asymptotic distributional behaviour of transformations preserving infinite measures. J. Anal. Math. 39 (1981), 203234.CrossRefGoogle Scholar
Eagleson, G. K.. Some simple conditions for limit theorems to be mixing. Teor. Veroyatn. Primen. 21 (1976), 653660.Google Scholar
Korepanov, A.. Equidistribution for nonuniformly expanding dynamical systems, and application to the almost sure invariance principle. Comm. Math. Phys. 359(3) (2018), 11231138.CrossRefGoogle Scholar
Melbourne, I. and Nicol, M.. Almost sure invariance principle for nonuniformly hyperbolic systems. Comm. Math. Phys. 260 (2005), 131146.CrossRefGoogle Scholar
Zweimüller, R.. Mixing limit theorems for ergodic transformations. J. Theoret. Probab. 20 (2007), 10591071.CrossRefGoogle Scholar

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