{
  "id": "2107.12860",
  "version": "v1",
  "published": "2021-07-27T14:49:07.000Z",
  "updated": "2021-07-27T14:49:07.000Z",
  "title": "The large deviation behavior of lacunary sums",
  "authors": [
    "Lorenz Fruehwirth",
    "Joscha Prochno",
    "Michael Juhos"
  ],
  "categories": [
    "math.PR",
    "math.NT"
  ],
  "abstract": "We study the large deviation behavior of lacunary sums $(S_n/n)_{n\\in \\mathbb{N} }$ with $S_n:= \\sum_{k=1}^n f(a_kU)$, $n\\in\\mathbb{N}$, where $U$ is uniformly distributed on $[0,1]$, $(a_k)_{k\\in\\mathbb{N}}$ is an Hadamard gap sequence, and $f\\colon \\mathbb{R}\\to \\mathbb{R} $ is a $1$-periodic, (Lipschitz-)continuous mapping. In the case of large gaps, we show that the normalized partial sums satisfy a large deviation principle at speed $n$ and with a good rate function which is the same as in the case of independent and identically distributed random variables $U_k$, $k\\in\\mathbb{N}$, having uniform distribution on $[0,1]$. When the lacunary sequence $(a_k)_{k\\in\\mathbb{N}}$ is a geometric progression, then we also obtain large deviation principles at speed $n$, but with a good rate function that is different from the independent case, its form depending in a subtle way on the interplay between the function $f$ and the arithmetic properties of the gap sequence. Our work generalizes some results recently obtained by Aistleitner, Gantert, Kabluchko, Prochno, and Ramanan [Large deviation principles for lacunary sums, preprint, 2020] who initiated this line of research for the case of lacunary trigonometric sums.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-27T14:49:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42A55",
      "60F10",
      "11L03",
      "37A05",
      "11D45",
      "11K70"
    ],
    "keywords": [
      "large deviation behavior",
      "lacunary sums",
      "large deviation principle",
      "rate function",
      "lacunary trigonometric sums"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}