{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:GFBKXR45GAYLUCQI5RWOZBZ3OY","short_pith_number":"pith:GFBKXR45","schema_version":"1.0","canonical_sha256":"3142abc79d3030ba0a08ec6cec873b761897a72ace20ea62b51218f76baef959","source":{"kind":"arxiv","id":"1602.05565","version":2},"attestation_state":"computed","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 $n, d \\rightarrow \\infty$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1602.05565","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-02-17T20:43:16Z","cross_cats_sorted":[],"title_canon_sha256":"e593051f0b733cf0da2838923c70ae691eef70ed05ac9b6f2a524cfa2db8e278","abstract_canon_sha256":"d85fe45b7c3a040c9fb176facb94b0e924316a11c95e66b6ff303ab21ccba22e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:39:47.929869Z","signature_b64":"77lIt/NWKrKH2EURXr4DqtFMjzC2FFCXNGlHF3RDqAj+Lgv8pPc4YwYZqH493OLEoZFAXYNxn85wGS4nqAUqAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3142abc79d3030ba0a08ec6cec873b761897a72ace20ea62b51218f76baef959","last_reissued_at":"2026-05-18T00:39:47.929084Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:39:47.929084Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1602.05565","created_at":"2026-05-18T00:39:47.929200+00:00"},{"alias_kind":"arxiv_version","alias_value":"1602.05565v2","created_at":"2026-05-18T00:39:47.929200+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.05565","created_at":"2026-05-18T00:39:47.929200+00:00"},{"alias_kind":"pith_short_12","alias_value":"GFBKXR45GAYL","created_at":"2026-05-18T12:30:19.053100+00:00"},{"alias_kind":"pith_short_16","alias_value":"GFBKXR45GAYLUCQI","created_at":"2026-05-18T12:30:19.053100+00:00"},{"alias_kind":"pith_short_8","alias_value":"GFBKXR45","created_at":"2026-05-18T12:30:19.053100+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2605.00229","citing_title":"A unified perspective on fine-tuning and sampling with diffusion and flow models","ref_index":187,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/GFBKXR45GAYLUCQI5RWOZBZ3OY","json":"https://pith.science/pith/GFBKXR45GAYLUCQI5RWOZBZ3OY.json","graph_json":"https://pith.science/api/pith-number/GFBKXR45GAYLUCQI5RWOZBZ3OY/graph.json","events_json":"https://pith.science/api/pith-number/GFBKXR45GAYLUCQI5RWOZBZ3OY/events.json","paper":"https://pith.science/paper/GFBKXR45"},"agent_actions":{"view_html":"https://pith.science/pith/GFBKXR45GAYLUCQI5RWOZBZ3OY","download_json":"https://pith.science/pith/GFBKXR45GAYLUCQI5RWOZBZ3OY.json","view_paper":"https://pith.science/paper/GFBKXR45","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1602.05565&json=true","fetch_graph":"https://pith.science/api/pith-number/GFBKXR45GAYLUCQI5RWOZBZ3OY/graph.json","fetch_events":"https://pith.science/api/pith-number/GFBKXR45GAYLUCQI5RWOZBZ3OY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GFBKXR45GAYLUCQI5RWOZBZ3OY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GFBKXR45GAYLUCQI5RWOZBZ3OY/action/storage_attestation","attest_author":"https://pith.science/pith/GFBKXR45GAYLUCQI5RWOZBZ3OY/action/author_attestation","sign_citation":"https://pith.science/pith/GFBKXR45GAYLUCQI5RWOZBZ3OY/action/citation_signature","submit_replication":"https://pith.science/pith/GFBKXR45GAYLUCQI5RWOZBZ3OY/action/replication_record"}},"created_at":"2026-05-18T00:39:47.929200+00:00","updated_at":"2026-05-18T00:39:47.929200+00:00"}