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Let $Y_n(\\mathbf {t}_1,\\mathbf {t}_2,\\sigma)=\\sqrt{p}({\\mathbf {x}}_n(\\mathbf {t}_1)^*(S_n+\\sigma I)^{-1}{\\mathbf {x}}_n(\\mathbf {t}_2)-{\\mathbf {x}}_n(\\mathbf {t}_1)^*{\\mathbf {x}}_n(\\mathbf {t}_2)m_n(\\sigma))$ in which $\\sigma>0$ and $m_n(\\sigma)=\\int\\frac{dF_{y_n}(x)}{x+\\sigma}$ where $F_{y_n}(x)$ is the Mar\\v{c}enko--Pastur law with parameter $y_n=p/n$; which converges to a positive constant as $n\\to\\infty$, and ${\\mathbf {x}}_n(\\mathbf {t}_1)$"},"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":"1201.0086","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-12-30T09:12:40Z","cross_cats_sorted":[],"title_canon_sha256":"7ba68f5d8a0c2ebd8a0f9b224bd45c878332f56ad2b852ea805142e98ab3da54","abstract_canon_sha256":"05624e2097cde92c3bac9067fbb15716fc7f9e56b7cd4524dd25a572b7a5e0d3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:05:26.268903Z","signature_b64":"Fy++WkBn36M5Wt44Z7GrZkRMHnXvhGg3sfMSXY1buE0NN4DiqhnkppIYTc0h2oW8WZcjP5CN0sWvEFT7UU6/Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2992d8f53bd2ffddf79371434c51fe6149878e32488aa57702e0e725cebec6b3","last_reissued_at":"2026-05-18T04:05:26.268223Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:05:26.268223Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotic properties of eigenmatrices of a large sample covariance matrix","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"H. X. Liu, W. K. Wong, Z. D. Bai","submitted_at":"2011-12-30T09:12:40Z","abstract_excerpt":"Let $S_n=\\frac{1}{n}X_nX_n^*$ where $X_n=\\{X_{ij}\\}$ is a $p\\times n$ matrix with i.i.d. complex standardized entries having finite fourth moments. Let $Y_n(\\mathbf {t}_1,\\mathbf {t}_2,\\sigma)=\\sqrt{p}({\\mathbf {x}}_n(\\mathbf {t}_1)^*(S_n+\\sigma I)^{-1}{\\mathbf {x}}_n(\\mathbf {t}_2)-{\\mathbf {x}}_n(\\mathbf {t}_1)^*{\\mathbf {x}}_n(\\mathbf {t}_2)m_n(\\sigma))$ in which $\\sigma>0$ and $m_n(\\sigma)=\\int\\frac{dF_{y_n}(x)}{x+\\sigma}$ where $F_{y_n}(x)$ is the Mar\\v{c}enko--Pastur law with parameter $y_n=p/n$; which converges to a positive constant as $n\\to\\infty$, and ${\\mathbf {x}}_n(\\mathbf {t}_1)$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.0086","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":""},"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":"1201.0086","created_at":"2026-05-18T04:05:26.268377+00:00"},{"alias_kind":"arxiv_version","alias_value":"1201.0086v1","created_at":"2026-05-18T04:05:26.268377+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.0086","created_at":"2026-05-18T04:05:26.268377+00:00"},{"alias_kind":"pith_short_12","alias_value":"FGJNR5J32L75","created_at":"2026-05-18T12:26:28.662955+00:00"},{"alias_kind":"pith_short_16","alias_value":"FGJNR5J32L75354T","created_at":"2026-05-18T12:26:28.662955+00:00"},{"alias_kind":"pith_short_8","alias_value":"FGJNR5J3","created_at":"2026-05-18T12:26:28.662955+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/FGJNR5J32L75354TOFBUYUP6MF","json":"https://pith.science/pith/FGJNR5J32L75354TOFBUYUP6MF.json","graph_json":"https://pith.science/api/pith-number/FGJNR5J32L75354TOFBUYUP6MF/graph.json","events_json":"https://pith.science/api/pith-number/FGJNR5J32L75354TOFBUYUP6MF/events.json","paper":"https://pith.science/paper/FGJNR5J3"},"agent_actions":{"view_html":"https://pith.science/pith/FGJNR5J32L75354TOFBUYUP6MF","download_json":"https://pith.science/pith/FGJNR5J32L75354TOFBUYUP6MF.json","view_paper":"https://pith.science/paper/FGJNR5J3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1201.0086&json=true","fetch_graph":"https://pith.science/api/pith-number/FGJNR5J32L75354TOFBUYUP6MF/graph.json","fetch_events":"https://pith.science/api/pith-number/FGJNR5J32L75354TOFBUYUP6MF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FGJNR5J32L75354TOFBUYUP6MF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FGJNR5J32L75354TOFBUYUP6MF/action/storage_attestation","attest_author":"https://pith.science/pith/FGJNR5J32L75354TOFBUYUP6MF/action/author_attestation","sign_citation":"https://pith.science/pith/FGJNR5J32L75354TOFBUYUP6MF/action/citation_signature","submit_replication":"https://pith.science/pith/FGJNR5J32L75354TOFBUYUP6MF/action/replication_record"}},"created_at":"2026-05-18T04:05:26.268377+00:00","updated_at":"2026-05-18T04:05:26.268377+00:00"}