{
  "id": "2210.08157",
  "version": "v1",
  "published": "2022-10-15T01:14:42.000Z",
  "updated": "2022-10-15T01:14:42.000Z",
  "title": "Central limit theorem and Berry-Esseen bounds for a branching random walk with immigration in a random environment",
  "authors": [
    "Chunmao Huang",
    "Yukun Ren",
    "Runze Li"
  ],
  "comment": "24 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider a branching random walk on $d$-dimensional real space with immigration in a time-dependent random environment. Let $Z_n(\\mathbf t)$ be the so-called partition function of the process, namely, the moment generating function of the counting measure describing the dispersion of individuals at time $n$. For $\\mathbf t$ fixed, the logarithm $\\log Z_n(\\mathbf t)$ satisfies a central limit theorem. By studying the logarithmic moments of the intrinsic submartingale of the system and its convergence rates, we establish the uniform and non-uniform Berry-Esseen bounds corresponding to the central limit theorem, and discover the exact convergence rate in the central limit theorem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-15T01:14:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "central limit theorem",
      "branching random walk",
      "immigration",
      "dimensional real space",
      "exact convergence rate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}