{"paper":{"title":"A high-dimensional CLT in $\\mathcal{W}_2$ distance with near optimal convergence rate","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Alex Zhai","submitted_at":"2016-02-17T20:43:16Z","abstract_excerpt":"Let $X_1, \\ldots , X_n$ be i.i.d. random vectors in $\\mathbb{R}^d$ with $\\|X_1\\| \\le \\beta$. Then, we show that $\\frac{1}{\\sqrt{n}}(X_1 + \\ldots + X_n)$ converges to a Gaussian in quadratic transportation (also known as \"Kantorovich\" or \"Wasserstein\") distance at a rate of $O\\left( \\frac{\\sqrt{d} \\beta \\log n}{\\sqrt{n}} \\right)$, improving a result of Valiant and Valiant. The main feature of our theorem is that the rate of convergence is within $\\log n$ of optimal for $n, d \\rightarrow \\infty$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.05565","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}