{
  "id": "1109.0635",
  "version": "v3",
  "published": "2011-09-03T16:42:22.000Z",
  "updated": "2014-12-02T07:14:03.000Z",
  "title": "Limit theorems for von Mises statistics of a measure preserving transformation",
  "authors": [
    "Manfred Denker",
    "Mikhail Gordin"
  ],
  "journal": "Probability and Related Fields 160 (2014), 1-45",
  "doi": "10.1007/s00440-013-0522-z",
  "categories": [
    "math.DS"
  ],
  "abstract": "For a measure preserving transformation $T$ of a probability space $(X,\\mathcal F,\\mu)$ we investigate almost sure and distributional convergence of random variables of the form $$x \\to \\frac{1}{C_n} \\sum_{i_1<n,...,i_d<n} f(T^{i_1}x,...,T^{i_d}x),\\, n=1,2,..., $$ where $f$ (called the \\emph{kernel}) is a function from $X^d$ to $\\R$ and $C_1, C_2,...$ are appropriate normalizing constants. We observe that the above random variables are well defined and belong to $L_r(\\mu)$ provided that the kernel is chosen from the projective tensor product $$L_p(X_1,\\mathcal F_1, \\mu_1) \\otimes_{\\pi}...\\otimes_{\\pi} L_p(X_d,\\mathcal F_d, \\mu_d)\\subset L_p(\\mu^d)$$ with $p=d\\,r,\\, r\\ \\in [1, \\infty).$ We establish a form of the individual ergodic theorem for such sequences. Next, we give a martingale approximation argument to derive a central limit theorem in the non-degenerate case (in the sense of the classical Hoeffding's decomposition). Furthermore, for $d=2$ and a wide class of canonical kernels $f$ we also show that the convergence holds in distribution towards a quadratic form $\\sum_{m=1}^{\\infty} \\lambda_m\\eta^2_m$ in independent standard Gaussian variables $\\eta_1, \\eta_2,...$. Our results on the distributional convergence use a $T$--\\,invariant filtration as a prerequisite and are derived from uni- and multivariate martingale approximations.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-02-24T15:20:26.000Z",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-12-02T07:14:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "measure preserving transformation",
      "von mises statistics",
      "limit theorem",
      "independent standard gaussian variables",
      "distributional convergence"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1109.0635D"
    }
  }
}