The Slide Dimension of Point Processes

Abstract

We associate with any finite subset of a metric space an infinite sequence of scale invariant numbers $\rho_1,\rho_2,\dots$ 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 $1/\rho_1$ converges to the Hausdorff dimension. We use the $\rho_n$ 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 $\rho_n$ and an explicit formula is derived for the calculation of $\rho_1$. For a uniform random variable X on $[0,1]^n$, evidence is given that $\rho_1(X) =1/n$ and $\rho_2(X) =-\pi^2/(6n^2)$ and simulations with a normal variable $Z$ suggest that $\rho_1(Z) =4/\pi$ and $\rho_2(Z) =-1$. Some potential applications to spatial statistics are considered.