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 mem- ory 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).

### 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 mem- ory 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).
##### Scheda breve Scheda completa Scheda completa (DC)
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11697/20204`
##### Citazioni
• ND
• ND
• ND