Journal of Theoretical Probability

, Volume 28, Issue 1, pp 137–183 | Cite as

Strong Invariance Principles with Rate for “Reverse” Martingale Differences and Applications

  • Christophe CunyEmail author
  • Florence Merlevède


In this paper, we obtain almost sure invariance principles with rate of order \(n^{1/p}\log ^\beta n\), \(2< p\le 4\), for sums associated with a sequence of reverse martingale differences. Then, we apply those results to obtain similar conclusions in the context of some non-invertible dynamical systems. For instance, we treat several classes of maps of the interval (for possibly unbounded observables) or smooth dynamical systems under a very weak regularity condition on the observables.


Expanding maps Smooth dynamical systems  Strong invariance principle Reverse martingale 

Mathematics Subject Classification (2010)

37E05 37C30 60F15 



The authors would like to thank the two referees for carefully reading the manuscript and for numerous suggestions that improved the presentation of this paper.


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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Laboratoire MAS, Ecole Centrale de ParisGrande Voie des VignesChatenay-Malabry CedexFrance
  2. 2.Université Paris Est, LAMA, CNRS UMR 8050Champs-Sur-MarneFrance

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