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Journal of Statistical Physics

, Volume 148, Issue 3, pp 548–564 | Cite as

Fluctuation Bounds for Chaos Plus Noise in Dynamical Systems

  • Cesar MaldonadoEmail author
Article

Abstract

We are interested in time series of the form y n =x n +ξ n where {x n } is generated by a chaotic dynamical system and where ξ n models observational noise. Using concentration inequalities, we derive fluctuation bounds for the auto-covariance function, the empirical measure, the kernel density estimator and the correlation dimension evaluated along y 0,…,y n , for all n. The chaotic systems we consider include for instance the Hénon attractor for Benedicks-Carleson parameters.

Keywords

Fluctuation bounds Concentration inequalities Empirical estimators Observational noise Non uniformly hyperbolic dynamical systems 

Notes

Acknowledgements

The author acknowledges J.-R. Chazottes, P. Collet and the anonymous reviewers for the careful reading of the manuscript, for their suggestions and corrections. The author thanks S. Galatolo and the DMA, Pisa, Italy, for their warm hospitality and where part of this work was done. The author is infinitely indebted to Adriana Aguilar Hervert.

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.CPhTCNRS-École PolytechniquePalaiseau CedexFrance

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