{
  "abstract": "We investigate the optimal dimension growth rates for the high-dimensional Central Limit Theorem (CLT) over half spaces, Euclidean balls and Borel convex sets, under the existence of fourth moments. It has been observed in the literature that $d=o(n^c)$ with $0< c\\leq 1/2$ is sufficient for the CLT to hold over Euclidean balls. As a consequence, the CLT over half spaces should ideally hold under similar dimension growth rates. In case of Borel convex sets, however $d= o(n^{c})$ with $0< c\\leq 2/5$ is generally sufficient. The most recent results for CLT over Euclidean balls and Borel convex sets with respective rates $d= o(\\sqrt{n})$ and $d= o(n^{2/5})$ is due to Fang and Koike (2024) and under the existence of fourth moments. In this paper, we investigate whether these sufficient growth rates of $d$ with respect to $n$ are also necessary. Thus, our results completely characterize the sharpness of dimension growth rates under existence of fourth moments across half-spaces, Euclidean balls, and Borel convex sets.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24183",
  "author_total": 2,
  "authors": [
    {
      "name": "Mayukh Choudhury"
    },
    {
      "name": "Debraj Das"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7dkecnk1e597y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24183",
  "fulltext_url": null,
  "id": "7dkecnk1e597y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24183",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Probability and Statistical Research"
    ]
  ],
  "openalex": "W7222998892",
  "pdf_url": "https://arxiv.org/pdf/2610.24183",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7dkecnk1e597y.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24183",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24183"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Optimal dimension growth rates in the high dimensional Central Limit Theorems",
  "title_text": "Optimal dimension growth rates in the high dimensional Central Limit Theorems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/7dkecnk1e597y",
  "venue": "arXiv",
  "versions": []
}