MPEJ Volume 9, No. 5, 17 pp.
Received: Aug 11, 2003. Revised: Nov 5, 2003. Accepted: Nov 7, 2003.
D.Gabrielli, A.Galves, D.Guiol
Fluctuations of the Empirical Entropies of a Chain of Infinite Order
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).
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