The Bernoulli structure of discrete distributions
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912075989647360 |
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| author | Fontana, Roberto Semeraro, Patrizia |
| author_facet | Fontana, Roberto Semeraro, Patrizia |
| contents | Any discrete distribution with support on $\{0,\ldots, d\}$ can be constructed as the distribution of sums of Bernoulli variables. We prove that the class of $d$-dimensional Bernoulli variables $\boldsymbol{X}=(X_1,\ldots, X_d)$ whose sums $\sum_{i=1}^dX_i$ have the same distribution $p$ is a convex polytope $\mathcal{P}(p)$ and we analytically find its extremal points. Our main result is to prove that the Hausdorff measure of the polytopes $\mathcal{P}(p), p\in \mathcal{D}_d,$ is a continuous function $l(p)$ over $\mathcal{D}_d$ and it is the density of a finite measure $μ_s$ on $\mathcal{D}_d$ that is Hausdorff absolutely continuous. We also prove that the measure $μ_s$ normalized over the simplex $\mathcal{D}$ belongs to the class of Dirichlet distributions. We observe that the symmetric binomial distribution is the mean of the Dirichlet distribution on $\mathcal{D}$ and that when $d$ increases it converges to the mode. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_13920 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Bernoulli structure of discrete distributions Fontana, Roberto Semeraro, Patrizia Probability Statistics Theory Any discrete distribution with support on $\{0,\ldots, d\}$ can be constructed as the distribution of sums of Bernoulli variables. We prove that the class of $d$-dimensional Bernoulli variables $\boldsymbol{X}=(X_1,\ldots, X_d)$ whose sums $\sum_{i=1}^dX_i$ have the same distribution $p$ is a convex polytope $\mathcal{P}(p)$ and we analytically find its extremal points. Our main result is to prove that the Hausdorff measure of the polytopes $\mathcal{P}(p), p\in \mathcal{D}_d,$ is a continuous function $l(p)$ over $\mathcal{D}_d$ and it is the density of a finite measure $μ_s$ on $\mathcal{D}_d$ that is Hausdorff absolutely continuous. We also prove that the measure $μ_s$ normalized over the simplex $\mathcal{D}$ belongs to the class of Dirichlet distributions. We observe that the symmetric binomial distribution is the mean of the Dirichlet distribution on $\mathcal{D}$ and that when $d$ increases it converges to the mode. |
| title | The Bernoulli structure of discrete distributions |
| topic | Probability Statistics Theory |
| url | https://arxiv.org/abs/2410.13920 |