Nearest-Neighbor Analysis of a Family of Fractal Distributions

Abstract

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.