  We prove asymptotic results for a modification of the cross-entropy estimator
originally introduced by Ziv and Merhav in the Markovian setting in 1993. Our
results concern a more general class of decoupled measures on shift spaces over
a finite alphabet and in particular imply strong asymptotic consistency of the
modified estimator for all pairs of functions of stationary, irreducible,
finite-state Markov chains satisfying a mild decay condition. Our approach is
based on the study of a rescaled cumulant-generating function called the
cross-entropic pressure, importing to information theory some techniques from
the study of large deviations within the thermodynamic formalism.
