{
  "abstract": "Let $\\{X_i\\}_{i=1}^n$ be independent, identically distributed lattice random variables with zero mean, unit variance, and maximal span $h>0$. Let $S_n=(X_1+\\cdots+X_n)/\\sqrt n$. Let $Z_n$ denote the quantization of a standard Gaussian random variable on the support lattice of $S_n$. We show that the Rényi divergence $D_α(S_n\\|Z_n)$ of order $1<α<\\infty$ tends to 0 as $n\\to\\infty$ if and only if $D_α(S_{n_0}\\| Z_{n_0})<\\infty$ for some integer $n_0$ and a sub-Gaussian condition holds. Furthermore, we derive an Edgeworth-type asymptotic expansion of the divergence to arbitrary order. We also show that the Rényi divergence $D_\\infty(S_n\\|Z_n)$ of infinite order tends to 0 as $n\\to\\infty$ if and only if a sub-Gaussian condition, a curvature condition, and a uniform separation condition hold. These results constitute lattice counterparts of Rényi entropic central limit theorems for continuous random variables obtained by Bobkov, Chistyakov and Götze (\\emph{Ann. Probab.} \\textbf{47} (2019), 270--323) and Bobkov and Götze (\\emph{Ann. Probab.} \\textbf{53} (2025), 453--477). Uniform separation is an additional assumption beyond the sub-Gaussian and curvature conditions used in the continuous setting. The failure of uniform separation may cause an exponentially tilted version of $X_1$ to concentrate on a sublattice of max span $kh$ for some integer $k>1$, which yields the lower bound $D_\\infty(S_n\\|Z_n)\\ge\\log k$, and hence a failure of the entropic central limit theorem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.16144",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhen Fu"
    },
    {
      "name": "Jiange Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/56qq5r6kpw59g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.16144",
  "fulltext_url": null,
  "id": "56qq5r6kpw59g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.16144",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Probability and Statistical Research"
    ]
  ],
  "openalex": "W7203711710",
  "pdf_url": "https://arxiv.org/pdf/2608.16144",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/56qq5r6kpw59g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2608.16144",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.16144"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Central limit theorem in Rényi divergence for lattice random variables",
  "title_text": "Central limit theorem in Rényi divergence for lattice random variables",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/56qq5r6kpw59g",
  "venue": "arXiv",
  "versions": []
}