The Bernoulli structure of discrete distributions

2024/10/17 by Roberto Fontana, Fontana, Roberto, Patrizia Semeraro +1 · 1 citation
Computer Science · Decision Sciences · Mathematics · #Advanced Statistical Process Monitoring #FOS: Mathematics #Mathematical and Theoretical Analysis #Probability (math.PR) #Statistical and Computational Modeling #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2410.13920

openalex publication_date 2024/10/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Any discrete distribution with support on \0,…, d\ can be constructed as the distribution of sums of Bernoulli variables. We prove that the class of d-dimensional Bernoulli variables \boldsymbolX=(X₁,…, X_d) whose sums ∑_i=1^dX_i have the same distribution p is a convex polytope P(p) and we analytically find its extremal points. Our main result is to prove that the Hausdorff measure of the polytopes P(p), p∈ D_d, is a continuous function l(p) over D_d and it is the density of a finite measure μ_s on D_d that is Hausdorff absolutely continuous. We also prove that the measure μ_s normalized over the simplex D belongs to the class of Dirichlet distributions. We observe that the symmetric binomial distribution is the mean of the Dirichlet distribution on D and that when d increases it converges to the mode.

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• Geometrical representation and dependence structure of three-dimensional Bernoulli distributions

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