The Slide Dimension of Point Processes

Abstract

We associate with any finite subset of a metric space an infinite sequence of scale invariant numbers $ρ_1,ρ_2,\dots$ 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 $ρ_n$ 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 $ρ_n$ 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.