{
  "id": "2404.16756",
  "version": "v1",
  "published": "2024-04-25T17:12:09.000Z",
  "updated": "2024-04-25T17:12:09.000Z",
  "title": "Concentration inequalities for Poisson $U$-statistics",
  "authors": [
    "Gilles Bonnet",
    "Anna Gusakova"
  ],
  "comment": "Preliminary version",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this article we obtain concentration inequalities for Poisson $U$-statistics $F_m(f,\\eta)$ of order $m\\ge 1$ with kernels $f$ under general assumptions on $f$ and the intensity measure $\\gamma \\Lambda$ of underlying Poisson point process $\\eta$. The main result are new concentration bounds of the form \\[ \\mathbb{P}(|F_m ( f , \\eta) -\\mathbb{E} F_m ( f , \\eta)| \\ge t)\\leq 2\\exp(-I(\\gamma,t)), \\] where $I(\\gamma,t)$ satisfies $I(\\gamma,t)=\\Theta(t^{1\\over m}\\log t)$ as $t\\to\\infty$ and $\\gamma$ is fixed. The function $I(\\gamma,t)$ is given explicitly in terms of parameters of the assumptions satisfied by $f$ and $\\Lambda$. One of the key ingredients of the proof are fine bounds for the centred moments of $F_m(f,\\eta)$. We discuss the optimality of obtained bounds and consider a number of applications related to Gilbert graphs and Poisson hyperplane processes in constant curvature spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-25T17:12:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "concentration inequalities",
      "statistics",
      "poisson point process",
      "poisson hyperplane processes",
      "constant curvature spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}