Orthogonal Dice
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2020
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| _version_ | 1866914009602588672 |
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| author | Bastian, Caleb Deen Rabitz, Herschel Rempala, Grzegorz A |
| author_facet | Bastian, Caleb Deen Rabitz, Herschel Rempala, Grzegorz A |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2009_10503 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Orthogonal Dice Bastian, Caleb Deen Rabitz, Herschel Rempala, Grzegorz A Probability 60G57, 60G55, 60F05 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. |
| title | Orthogonal Dice |
| topic | Probability 60G57, 60G55, 60F05 |
| url | https://arxiv.org/abs/2009.10503 |