  In this paper we extend the results of Lenci and Rey-Bellet on the large
deviation upper bound of the distribution measures of local Hamiltonians with
respect to a Gibbs state, in the setting of translation-invariant finite-range
interactions. We show that a certain factorization property of the reference
state is sufficient for a large deviation upper bound to hold and that this
factorization property is satisfied by Gibbs states of the above kind as well
as finitely correlated states. As an application of the methods the Chernoff
bound for correlated states with factorization property is studied. In the
specific case of the distributions of the ergodic averages of a one-site
observable with respect to an ergodic finitely correlated state the spectral
theory of positive maps is applied to prove the full large deviation principle.
