## Fluctuations of the empirical entropies of a chain of infinite order.(English)Zbl 1038.60097

By a chain of infinite order the authors mean a finite state strictly stationary process in which at each step the conditional probability of a state given the past does depend on the whole of it. The paper is concerned with the fluctuations of the empirical entropy of a chain of infinite order. Two possible definitions for the empirical entropy are considered, both based on the empirical distribution of cylinders with length $$c\log n$$, where $$n$$ is the sample size and $$c$$ a suitable constant. The first one is the conditional entropy of the empirical distribution, given a past with length growing logarithmically with the sample size. The second one is the rescaled entropy of the empirical distribution of the cylinders of size growing logarithmically with the sample size. Assuming an exponentially fast loss of memory (in a well-defined sense), the authors prove a central limit theorem in the first case and show that in the second case the empirical entropy does not have Gaussian fluctuations around the theoretical entropy. The last result solves a problem raised some 40 years ago by this reviewer [Ann. Math. Stat. 36, 1433–1436 (1965; Zbl 0158.36005)].

### MSC:

 60K99 Special processes 60F05 Central limit and other weak theorems 94A17 Measures of information, entropy

Zbl 0158.36005
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