Statistical estimation of measure invariants

Timofeev, E.

doi:10.1090/S1061-0022-06-00919-8

Statistical estimation of measure invariants
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by E. A. Timofeev
Translated by: A. Plotkin

St. Petersburg Math. J. 17 (2006), 527-551

DOI: https://doi.org/10.1090/S1061-0022-06-00919-8

Published electronically: March 21, 2006

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Abstract:

New invariants of measures, called the $\beta$-statentropy, are described. They are similar to the entropy and the $HP$-spectrum for dimensions. The $\beta$-statentropy admits construction of a statistical estimator calculated by $n$ independent points distributed in accordance with a given measure. The accuracy of this estimator is $\mathcal {O}(n^{-c})$, where $c$ is some constant, and the complexity of calculation is $\mathcal {O}(n^2)$.

It is shown that for an exact dimensional measure the $0$-statentropy coincides with the Hausdorff dimension, and for a Markov measure the $\beta$-statentropy coincides with the $HP$-spectrum for dimensions.

An application of the $\beta$-statentropy to finding the entropy and dimensional characteristics of dynamical systems is described.

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