Orthogonal Dice

Abstract

In this paper, we introduce a family of discrete rectangular uniform distributions on the natural numbers-referred to as orthogonal dice-characterized by the property that their means equal their variances. These distributions arise naturally in statistics and applied mathematics. We show that the orthogonal dice correspond to solutions of a quadratic Diophantine equation on the naturals, exhibiting divisibility properties tied to their dimensions, generating coprime arithmetic progressions, yielding disjoint partitions of the naturals, and displaying self-similarity. Their associated random counting measures (mixed binomial processes) exhibit interesting structural properties, including orthogonal splitting and convergence to Poisson limits. As a result, the orthogonal dice define canonical stochastic processes that that may be used to construct Brownian and geometric Brownian motions. More broadly, they serve as Poisson-like building blocks-natural substrates for modeling systems with bounded counts. Furthermore, they induce a trichotomy within the broader class of such distributions, partitioning them into three infinite subfamilies-negative, orthogonal, and positive-according to their mean-variance relationships.