{
  "id": "1507.03721",
  "version": "v1",
  "published": "2015-07-14T04:59:06.000Z",
  "updated": "2015-07-14T04:59:06.000Z",
  "title": "Some measure-theoretic properties of U-statistics applied in statistical physics",
  "authors": [
    "Irina Navrotskaya"
  ],
  "categories": [
    "math.CA",
    "cond-mat.stat-mech",
    "math.PR"
  ],
  "abstract": "This paper investigates the relationship between various measure-theoretic properties of U-statistics with fixed sample size $N$ and the same properties of their kernels. Specifically, the random variables are replaced with elements in some measure space $(\\Lambda; dx)$, the resultant real-valued functions on $\\Lambda^N$ being called generalized $N$-means. It is shown that a.e. convergence of sequences, measurability, essential boundedness and, under certain conditions, integrability with respect to probability measures of generalized $N$-means and their kernels are equivalent. These results are crucial for the solution of the inverse problem in classical statistical mechanics in the canonical formulation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-07-14T04:59:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A20",
      "28A25",
      "28A35",
      "62H12",
      "82B21"
    ],
    "keywords": [
      "measure-theoretic properties",
      "statistical physics",
      "u-statistics",
      "inverse problem",
      "probability measures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150703721N"
    }
  }
}