{"paper":{"title":"Central Limit Theorem in High Dimensions : The Optimal Bound on Dimension Growth Rate","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Debraj Das, Soumendra Lahiri","submitted_at":"2020-08-10T19:53:57Z","abstract_excerpt":"In this article, we try to give an answer to the simple question: ``\\textit{What is the critical growth rate of the dimension $p$ as a function of the sample size $n$ for which the Central Limit Theorem holds uniformly over the collection of $p$-dimensional hyper-rectangles ?''}. Specifically, we are interested in the normal approximation of suitably scaled versions of the sum $\\sum_{i=1}^{n}X_i$ in $\\mathcal{R}^p$ uniformly over the class of hyper-rectangles $\\mathcal{A}^{re}=\\{\\prod_{j=1}^{p}[a_j,b_j]\\cap\\mathcal{R}:-\\infty\\leq a_j\\leq b_j \\leq \\infty, j=1,\\ldots,p\\}$, where $X_1,\\dots,X_n$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2008.04389","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2008.04389/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}