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

Statistical consequences of the Devroye inequality for processes. Applications to a class of non-uniformly hyperbolic dynamical systems