{
  "id": "2102.05094",
  "version": "v1",
  "published": "2021-02-09T19:48:16.000Z",
  "updated": "2021-02-09T19:48:16.000Z",
  "title": "A probabilistic approach to the Erdös-Kac theorem for additive functions",
  "authors": [
    "Louis H. Y. Chen",
    "Arturo Jaramillo",
    "Xiaochuan Yang"
  ],
  "categories": [
    "math.PR",
    "math.NT"
  ],
  "abstract": "We present a new perspective of assessing the rates of convergence to the Gaussian and Poisson distributions in the Erd\\\"os-Kac theorem for additive arithmetic functions $\\psi$ of a random integer $J_n$ uniformly distributed over $\\{1,...,n\\}$. Our approach is probabilistic, working directly on spaces of random variables without any use of Fourier analytic methods, and our $\\psi$ is more general than those considered in the literature. Our main results are (i) bounds on the Kolmogorov distance and Wasserstein distance between the distribution of the normalized $\\psi(J_n)$ and the standard Gaussian distribution, and (ii) bounds on the Kolmogorov distance and total variation distance between the distribution of $\\psi(J_n)$ and a Poisson distribution under mild additional assumptions on $\\psi$. Our results generalize the existing ones in the literature.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-09T19:48:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "62E17",
      "60F05",
      "11N60",
      "11K65"
    ],
    "keywords": [
      "probabilistic approach",
      "erdös-kac theorem",
      "additive functions",
      "kolmogorov distance",
      "poisson distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}