{
  "id": "2302.04539",
  "version": "v1",
  "published": "2023-02-09T10:11:33.000Z",
  "updated": "2023-02-09T10:11:33.000Z",
  "title": "Some remarks on the ergodic theorem for $U$-statistics",
  "authors": [
    "Davide Giraudo",
    "Dalibor Volny",
    "Herold G. Dehling"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "In this note, we investigate the convergence of a $U$-statistic of order two having stationary ergodic data. We will find sufficient conditions for the almost sure and $L^1$ convergence and present some counter-examples showing that the $U$-statistic itself might fail to converge: centering is needed as well as boundedness of $\\sup_{j\\geq 2}\\mathbb{E}[|h(X_1,X_j)|]$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-09T10:11:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ergodic theorem",
      "stationary ergodic data",
      "convergence",
      "sufficient conditions",
      "boundedness"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}