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

摘要

In this paper, we apply Devroye inequality to study various statisticalestimators and fluctuations of observables for processes. Most of theseobservables are suggested by dynamical systems. These applications concern theco-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-uniformlyhyperbolic dynamical systems introduced in \cite{young}. In the second appendixwe prove that, if the decay of correlations holds with a common rate for allpairs of functions, then it holds uniformly in the function spaces. In the lastappendix we prove that for the subclass of one-dimensional systems studied in\cite{young} the density of the absolutely continuous invariant measure belongsto a Besov space.