{
  "id": "2408.10896",
  "version": "v1",
  "published": "2024-08-20T14:33:36.000Z",
  "updated": "2024-08-20T14:33:36.000Z",
  "title": "Monotonicity equivalence and synchronizability for a system of probability distributions",
  "authors": [
    "Motoya Machida"
  ],
  "comment": "25 pages, 6 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "A system $(P_\\alpha: \\alpha\\in\\mathcal{A})$ of probability distributions on a partially ordered set (poset) $\\mathcal{S}$ indexed by another poset $\\mathcal{A}$ can be realized by a system of $\\mathcal{S}$-valued random variables $X_\\alpha$'s marginally distributed as $P_\\alpha$. It is called realizably monotone if $X_\\alpha\\le X_\\beta$ in $\\mathcal{S}$ whenever $\\alpha\\le\\beta$ in $\\mathcal{A}$. Such a system necessarily is stochastically monotone, that is, it satisfies $P_\\alpha\\preceq P_\\beta$ in stochastic ordering whenever $\\alpha \\le \\beta$. It has been known exactly when these notions of monotonicity are equivalent except for a certain subclass of acyclic posets, called Class W. In this paper we introduce inverse probability transforms and synchronizing bijections recursively when $\\mathcal{S}$ is a poset of Class W and $\\mathcal{A}$ is synchronizable, and validate monotonicity equivalence by constructing $(X_\\alpha: \\alpha\\in\\mathcal{A})$ explicitly. We also show that synchronizability is necessary for monotonicity equivalence when $\\mathcal{S}$ is in Class W.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-20T14:33:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E05",
      "06A06",
      "05C05",
      "05C38"
    ],
    "keywords": [
      "probability distributions",
      "synchronizability",
      "validate monotonicity equivalence",
      "inverse probability transforms",
      "acyclic posets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}