{
  "id": "2410.13920",
  "version": "v1",
  "published": "2024-10-17T10:22:31.000Z",
  "updated": "2024-10-17T10:22:31.000Z",
  "title": "The Bernoulli structure of discrete distributions",
  "authors": [
    "Roberto Fontana",
    "Patrizia Semeraro"
  ],
  "categories": [
    "math.PR",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "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 $\\mu_s$ on $\\mathcal{D}_d$ that is Hausdorff absolutely continuous. We also prove that the measure $\\mu_s$ 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-17T10:22:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "discrete distribution",
      "bernoulli structure",
      "dirichlet distribution",
      "dimensional bernoulli variables",
      "symmetric binomial distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}