Gabrielli D., Galves A., Guiol D.
Fluctuations of the Empirical Entropies of a Chain of Infinite
Order.
(48K, LATeX)
ABSTRACT. 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 $c\log{n}$, 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).