{
  "id": "1902.05520",
  "version": "v1",
  "published": "2019-02-14T17:56:19.000Z",
  "updated": "2019-02-14T17:56:19.000Z",
  "title": "Generalized semimodularity: order statistics",
  "authors": [
    "Iosif Pinelis"
  ],
  "comment": "To appear in the proceedings of the conference High Dimensional Probability 8, held in Oaxaca (Mexico) in 2017",
  "categories": [
    "math.PR",
    "math.CO",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "A notion of generalized $n$-semimodularity is introduced, which extends that of (sub/super)mod\\-ularity in four ways at once. The main result of this paper, stating that every generalized $(n\\colon\\!2)$-semimodular function on the $n$th Cartesian power of a distributive lattice is generalized $n$-semimodular, may be considered a multi/infinite-dimensional analogue of the well-known Muirhead lemma in the theory of Schur majorization. This result is also similar to a discretized version of the well-known theorem due to Lorentz, which latter was given only for additive-type functions. Illustrations of our main result are presented for counts of combinations of faces of a polytope; one-sided potentials; multiadditive forms, including multilinear ones -- in particular, permanents of rectangular matrices and elementary symmetric functions; and association inequalities for order statistics. Based on an extension of the FKG inequality due to Rinott \\& Saks and Aharoni \\& Keich, applications to correlation inequalities for order statistics are given as well.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-14T17:56:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "06D99",
      "26D15",
      "26D20",
      "60E15",
      "05A20",
      "05B35",
      "06A07",
      "60C05",
      "62H05",
      "62H10",
      "82D99",
      "90C27"
    ],
    "keywords": [
      "order statistics",
      "generalized semimodularity",
      "main result",
      "th cartesian power",
      "elementary symmetric functions"
    ],
    "tags": [
      "conference paper"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}