In this paper, we apply the 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 the empirical measure, the shadowing property and the almost-sure central limit theorem. We proved (Chazottes et al 2005 Nonlinearity 18 2323–40) that the Devroye inequality holds for a class of non-uniformly hyperbolic dynamical systems introduced in Young (1998 Ann. Math. 147 585–650). 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 Young, the density of the absolutely continuous invariant measure belongs to a Besov space.