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Nearest-Neighbor Analysis of a Family of Fractal Distributions
Colleen D. Cutler and Donald A. Dawson
The Annals of Probability
Vol. 18, No. 1 (Jan., 1990), pp. 256-271
Published by: Institute of Mathematical Statistics
Stable URL: http://www.jstor.org/stable/2244237
Page Count: 16
You can always find the topics here!Topics: Hausdorff dimensions, Entropy, Fractals, Neighborhoods, Borel sets, Ergodic theory, Mathematical theorems, Random variables, Integers
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In this paper we use a central limit theorem for entropy due to Ibragimov to obtain limit theorems for linear normalizations of the log minimum distance when observations are sampled from measures belonging to a family of fractal distributions. It is shown that in almost all cases the limit distribution is Gaussian with parameters determined in part by the Hausdorff dimension associated with the underlying measure. Exceptions to this rule include absolutely continuous measures which obey the classical extreme value limit laws.
The Annals of Probability © 1990 Institute of Mathematical Statistics