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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
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Abstract

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

References

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