Statistical Consequences of Devroye Inequality for Processes. Applications to a Class of Non-Uniformly Hyperbolic Dynamical Systems

Details Statistical Consequences of Devroye Inequality for Processes. Applications to a Class of Non-Uniformly Hyperbolic Dynamical Systems Statistical Consequences of Devroye Inequality for Processes. Applications to a Class of Non-Uniformly Hyperbolic Dynamical Systems

2004-12-08

arxiv.org/abs/math/0412167v2

Mathematics

Dynamical Systems

33 pages; companion of the paper math.DS/0412166; corrected version; to appear in Nonlinearity

Scientific paper

10.1088/0951-7715/18/5/024

In this paper, we apply Devroye inequality to study various statistical estimators and fluctuations of observables for processes. Most of these observables are suggested by dynamical systems. These applications concern the co-variance function, the integrated periodogram, the correlation dimension, the kernel density estimator, the speed of convergence of empirical measure, the shadowing property and the almost-sure central limit theorem. We proved in \cite{CCS} that Devroye inequality holds for a class of non-uniformly hyperbolic dynamical systems introduced in \cite{young}. In the second appendix we prove that, if the decay of correlations holds with a common rate for all pairs of functions, then it holds uniformly in the function spaces. In the last appendix we prove that for the subclass of one-dimensional systems studied in \cite{young} the density of the absolutely continuous invariant measure belongs to a Besov space.

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