Orthogonal Dice

TL;DR

This paper introduces orthogonal dice, a family of discrete distributions with equal means and variances, revealing their mathematical properties, structural behaviors, and applications in stochastic process modeling.

Contribution

It characterizes orthogonal dice through quadratic Diophantine equations and explores their structural, divisibility, and stochastic properties, providing new tools for modeling bounded count systems.

Findings

Orthogonal dice satisfy a quadratic Diophantine equation.

They generate coprime arithmetic progressions and disjoint partitions of naturals.

Their associated processes converge to Poisson limits and can model Brownian motions.

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.