{
  "id": "1910.01439",
  "version": "v1",
  "published": "2019-10-03T13:09:27.000Z",
  "updated": "2019-10-03T13:09:27.000Z",
  "title": "Limit theorems for numbers of multiple returns in nonconventional arrays",
  "authors": [
    "Yuri Kifer"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "For a $\\psi$-mixing process $\\xi_0,\\xi_1,\\xi_2,...$ we consider the number $\\mathcal{N}_N$ of multiple returns $\\{\\xi_{q_{i,N}(n)}\\in\\Gamma_N,\\, i=1,...,\\ell\\}$ to a set $\\Gamma_N$ for $n$ until either a fixed number $N$ or until the moment $\\tau_N$ when another multiple return $\\{\\xi_{q_{i,N}(n)}\\in\\Delta_N,\\, i=1,...,\\ell\\}$ takes place for the first time where $\\Gamma_N\\cap\\Delta_N=\\emptyset$ and $q_{i,N},\\, i=1,...,\\ell$ are certain functions of $n$ taking on nonnegative integer values when $n$ runs from 0 to $N$. The dependence of $q_{i,N}(n)$'s on both $n$ and $N$ is the main novelty of the paper. Under some restrictions on the functions $q_{i,N}$ we obtain Poisson distributions limits of $\\mathcal{N}_N$ when counting is until $N$ as $N\\to\\infty$ and geometric distributions limits when counting is until $\\tau_N$ as $N\\to\\infty$. We obtain also similar results in the dynamical systems setup considering a $\\psi$-mixing shift $T$ on a sequence space $\\Omega$ and studying the number of multiple returns $\\{ T^{q_{i,N}(n)}\\omega\\in A^a_n,\\, i=1,...,\\ell\\}$ until the first occurrence of another multiple return $\\{ T^{q_{i,N}(n)}\\omega\\in A^b_m,\\, i=1,...,\\ell\\}$ where $A^a_n,\\, A_m^b$ are cylinder sets of length $n$ and $m$ constructed by sequences $a,b\\in\\Omega$, respectively, and chosen so that their probabilities have the same order.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-03T13:09:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "multiple return",
      "limit theorems",
      "nonconventional arrays",
      "poisson distributions limits",
      "geometric distributions limits"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}