On the Ziv-Merhav theorem beyond Markovianity II: leveraging the thermodynamic formalism

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.