Open Access
January, 1992 Entropy and Prefixes
Paul C. Shields
Ann. Probab. 20(1): 403-409 (January, 1992). DOI: 10.1214/aop/1176989934


Grassberger suggested an interesting entropy estimator, namely, $\frac{n \log n}{\sum^n_{i=1} L^n_i},$ where $L^n_i$ is the shortest prefix of $x_i, x_{i+1},\ldots$, which is not a prefix of any other $x_j, x_{j+1},\ldots,$ for $j \leq n$. We show that this estimator is not consistent for the general ergodic process, although it is consistent for Markov chains. A weaker trimmed mean type result is proved for the general case, namely, given $\varepsilon > 0$, eventually almost surely all but an $\varepsilon$ fraction of the $L^n_i/\log n$ will be within $\varepsilon$ of $1/H$. A related Hausdorff dimension conjecture is shown to be false.


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Paul C. Shields. "Entropy and Prefixes." Ann. Probab. 20 (1) 403 - 409, January, 1992.


Published: January, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0765.28016
MathSciNet: MR1143428
Digital Object Identifier: 10.1214/aop/1176989934

Primary: 28D20
Secondary: 60F15

Keywords: Entropy , Hausdorff dimension

Rights: Copyright © 1992 Institute of Mathematical Statistics

Vol.20 • No. 1 • January, 1992
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