The Slide Dimension of Point Processes

Abstract

We associate with any finite subset of a metric space an infinite sequence of scale invariant numbers ρ 1 , ρ 2 , . . . derived from a variant of differential entropy called the genial entropy. As statistics for point processes, these numbers often appear to converge in simulations and we give examples where 1/ρ 1 converges to the Hausdorff dimension. We use the ρn to define a new notion of dimension called the slide dimension for a special class of point processes on metric spaces. The slide calculus is developed to define ρn and an explicit formula is derived for the calculation of ρ 1 . For a uniform random variable X on [0, 1] n , evidence is given that ρ 1 (X) = 1/n and ρ 2 (X) = −π 2 /(6n 2 ) and simulations with a normal variable Z suggest that ρ 1 (Z) = 4/π and ρ 2 (Z) = −1. Some potential applications to spatial statistics are considered.