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.

翻译：我们证明了原始由Ziv和Merhav在1993年针对马尔可夫设定提出的交叉熵估计量改进版本的渐近结果。我们的结果适用于有限字母表移位空间上一类更广泛的解耦测度，特别地，对于满足温和衰减条件的平稳、不可约、有限状态马尔可夫链的所有函数对，该改进估计量具有强渐近一致性。我们的研究方法基于对称为交叉熵压力的重缩放累积生成函数的研究，将热力学形式化框架中大偏差研究的若干技术引入信息论领域。