Fluctuations of the Empirical Entropies of a Chain of Infinite Order

This paper addresses the question of the fluctuations of the empirical entropy of a chain of infinite order. We assume that the chain takes values on a finite alphabet and loses memory exponentially fast. We consider two possible definitions for the empirical entropy, both based on the empirical distribution of cylinders with length clog(n), where n is the size of the sample and c is a suitable constant. The first one is the conditional entropy of the empirical distribution, given a past with length growing logarithmically with the size of the sample. The second one is the rescaled entropy of the empirical distribution of the cylinders of size growing logarithmically with the size of the sample. We prove a central limit theorem for the first one. We also prove that the second one does not have Gaussian fluctuations. This solves a problem formulated in Iosifescu (1965).