{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:TTEPZFXLYTPCCVMMF626DKENSU","short_pith_number":"pith:TTEPZFXL","schema_version":"1.0","canonical_sha256":"9cc8fc96ebc4de21558c2fb5e1a88d9500f74e07e6b3f040ffec560f1a641d73","source":{"kind":"arxiv","id":"1012.0294","version":3},"attestation_state":"computed","paper":{"title":"Sharp bounds on the rate of convergence of the empirical covariance matrix","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.PR","authors_text":"Alain Pajor, Alexander E. Litvak, Nicole Tomczak-Jaegermann, Rados{\\l}aw Adamczak","submitted_at":"2010-12-01T20:40:01Z","abstract_excerpt":"Let $X_1,..., X_N\\in\\R^n$ be independent centered random vectors with log-concave distribution and with the identity as covariance matrix. We show that with overwhelming probability at least $1 - 3 \\exp(-c\\sqrt{n}\\r)$ one has $\n  \\sup_{x\\in S^{n-1}} \\Big|\\frac{1/N}\\sum_{i=1}^N (|<X_i, x>|^2 - \\E|<X_i, x>|^2\\r)\\Big|\n  \\leq C \\sqrt{\\frac{n/N}},$ where $C$ is an absolute positive constant. This result is valid in a more general framework when the linear forms $(<X_i,x>)_{i\\leq N, x\\in S^{n-1}}$ and the Euclidean norms $(|X_i|/\\sqrt n)_{i\\leq N}$ exhibit uniformly a sub-exponential decay. As a con"},"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":"1012.0294","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-12-01T20:40:01Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"dd2ac38d78c5ccdc0d29bba2c179ef98252cd1833d4dc8c7458d753fad8a0fe4","abstract_canon_sha256":"a29a214ceca1ce504370ffcd89492f5fb7157c9b43c70c41773ebb800c7867ab"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:41:56.701859Z","signature_b64":"Ci70ipniw92JAxXJcODRej+JLLAgFHwOdM1NgziY47fnlkg0o+xbQBWjyxxFUHJf3ToJyWTxEcrDgtlCliAODA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9cc8fc96ebc4de21558c2fb5e1a88d9500f74e07e6b3f040ffec560f1a641d73","last_reissued_at":"2026-05-18T03:41:56.700947Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:41:56.700947Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sharp bounds on the rate of convergence of the empirical covariance matrix","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.PR","authors_text":"Alain Pajor, Alexander E. Litvak, Nicole Tomczak-Jaegermann, Rados{\\l}aw Adamczak","submitted_at":"2010-12-01T20:40:01Z","abstract_excerpt":"Let $X_1,..., X_N\\in\\R^n$ be independent centered random vectors with log-concave distribution and with the identity as covariance matrix. We show that with overwhelming probability at least $1 - 3 \\exp(-c\\sqrt{n}\\r)$ one has $\n  \\sup_{x\\in S^{n-1}} \\Big|\\frac{1/N}\\sum_{i=1}^N (|<X_i, x>|^2 - \\E|<X_i, x>|^2\\r)\\Big|\n  \\leq C \\sqrt{\\frac{n/N}},$ where $C$ is an absolute positive constant. This result is valid in a more general framework when the linear forms $(<X_i,x>)_{i\\leq N, x\\in S^{n-1}}$ and the Euclidean norms $(|X_i|/\\sqrt n)_{i\\leq N}$ exhibit uniformly a sub-exponential decay. As a con"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.0294","kind":"arxiv","version":3},"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":"1012.0294","created_at":"2026-05-18T03:41:56.701102+00:00"},{"alias_kind":"arxiv_version","alias_value":"1012.0294v3","created_at":"2026-05-18T03:41:56.701102+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.0294","created_at":"2026-05-18T03:41:56.701102+00:00"},{"alias_kind":"pith_short_12","alias_value":"TTEPZFXLYTPC","created_at":"2026-05-18T12:26:15.391820+00:00"},{"alias_kind":"pith_short_16","alias_value":"TTEPZFXLYTPCCVMM","created_at":"2026-05-18T12:26:15.391820+00:00"},{"alias_kind":"pith_short_8","alias_value":"TTEPZFXL","created_at":"2026-05-18T12:26:15.391820+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/TTEPZFXLYTPCCVMMF626DKENSU","json":"https://pith.science/pith/TTEPZFXLYTPCCVMMF626DKENSU.json","graph_json":"https://pith.science/api/pith-number/TTEPZFXLYTPCCVMMF626DKENSU/graph.json","events_json":"https://pith.science/api/pith-number/TTEPZFXLYTPCCVMMF626DKENSU/events.json","paper":"https://pith.science/paper/TTEPZFXL"},"agent_actions":{"view_html":"https://pith.science/pith/TTEPZFXLYTPCCVMMF626DKENSU","download_json":"https://pith.science/pith/TTEPZFXLYTPCCVMMF626DKENSU.json","view_paper":"https://pith.science/paper/TTEPZFXL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1012.0294&json=true","fetch_graph":"https://pith.science/api/pith-number/TTEPZFXLYTPCCVMMF626DKENSU/graph.json","fetch_events":"https://pith.science/api/pith-number/TTEPZFXLYTPCCVMMF626DKENSU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TTEPZFXLYTPCCVMMF626DKENSU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TTEPZFXLYTPCCVMMF626DKENSU/action/storage_attestation","attest_author":"https://pith.science/pith/TTEPZFXLYTPCCVMMF626DKENSU/action/author_attestation","sign_citation":"https://pith.science/pith/TTEPZFXLYTPCCVMMF626DKENSU/action/citation_signature","submit_replication":"https://pith.science/pith/TTEPZFXLYTPCCVMMF626DKENSU/action/replication_record"}},"created_at":"2026-05-18T03:41:56.701102+00:00","updated_at":"2026-05-18T03:41:56.701102+00:00"}