{
  "id": "1910.07908",
  "version": "v1",
  "published": "2019-10-16T15:06:57.000Z",
  "updated": "2019-10-16T15:06:57.000Z",
  "title": "Limit theorems for numbers of returns in arrays under $φ$-mixing",
  "authors": [
    "Yuri Kifer"
  ],
  "comment": "arXiv admin note: substantial text overlap with arXiv:1910.01439",
  "categories": [
    "math.DS"
  ],
  "abstract": "We consider a $\\phi$-mixing shift $T$ on a sequence space $\\Om$ and study the number $\\cN_N$ of returns $\\{ T^{q_N(n)}\\om\\in A^a_n\\}$ at times $q_N(n)$ to a cylinder $A^a_n$ constructed by a sequence $a\\in\\Om$ where $n$ runs either until a fixed integer $N$ or until a time $\\tau_N$ of the first return $\\{ T^{q_N(n)}\\om\\in A^b_m\\}$ to another cylinder $A^b_m$ constructed by $b\\in\\Om$. Here $q_N(n)$ are certain functions of $n$ taking on nonnegative integer values when $n$ runs from 0 to $N$ and the dependence on $N$ is the main generalization here which requires certain conditions under which we obtain Poisson distributions limits of $\\cN_N$ when counting is until $N$ as $N\\to\\infty$ and geometric distributions limits when counting is until $\\tau_N$ as $N\\to\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-16T15:06:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "limit theorems",
      "geometric distributions limits",
      "poisson distributions limits",
      "first return",
      "sequence space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}