{
  "id": "2308.11168",
  "version": "v1",
  "published": "2023-08-22T03:54:32.000Z",
  "updated": "2023-08-22T03:54:32.000Z",
  "title": "Discretized Normal Approximation of Sums of Locally Dependent Random Variables via Stein's Method",
  "authors": [
    "Zhonggen Su",
    "Xiaolin Wang"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\{X_{i}, i\\in J\\}$ be a family of locally dependent non-negative integer-valued random variables with finite expectation and variance. We consider the sum $W=\\sum_{i\\in J}X_i$ and establish general error upper bounds for the total variation distance $d_{TV}(W, Y^{d})$, where $Y^{d}$ is the discretized normal distribution. The major ingredient of the proof is to approximate $W$ by a three-parametric intermediate random variable $M$ based on Stein's method. As applications, we study in detail four well-known examples, which are counting vertices of all edges point inward, birthday problem, counting monochromatic edges in uniformly colored graphs, and triangles in the Erd\\H{o}s-R\\'{e}nyi random graph. Through delicate analysis and computations we obtain sharper upper error bounds than existing results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-22T03:54:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "locally dependent random variables",
      "discretized normal approximation",
      "steins method",
      "general error upper bounds",
      "non-negative integer-valued random variables"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}