  In this paper, we prove an inequality, which we call "Devroye inequality",
for a large class of non-uniformly hyperbolic dynamical systems (M,f). This
class, introduced by L.-S. Young, includes families of piece-wise hyperbolic
maps (Lozi-like maps), scattering billiards (e.g., planar Lorentz gas),
unimodal and H{\'e}non-like maps. 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 Holder continuous function of n variables. In particular, we
can deal with observables which are not Birkhoff averages. We will show in
\cite{CCS} some applications of Devroye inequality to statistical properties of
this class of dynamical systems.
