Devroye inequality for a class of non-uniformly hyperbolic dynamical systems

In this paper we prove an inequality which we call the 'Devroye inequality' for a large class of non-uniformly hyperbolic dynamical systems (M, f). This class, introduced by Young, includes families of piecewise hyperbolic maps (Lozi-like maps), scattering billiards (e.g. planar Lorentz gas), unimodal and Hénon-like maps. The Devroye inequality provides an upper bound for the variance of observables of the form K(x, f(x), ..., f^n-1(x)), where K is any separately Hölder continuous function of n variables. In particular, we can deal with observables which are not Birkhoff averages. We will show in Chazottes et al (2005 Nonlinearity 18 2341-64) some applications of Devroye inequality to statistical properties of this class of dynamical systems.