{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:DYM7WXZV3GBPUW6KSST5ABR3OK","short_pith_number":"pith:DYM7WXZV","schema_version":"1.0","canonical_sha256":"1e19fb5f35d982fa5bca94a7d0063b72a7723458b51cbcd19a43f67deab30250","source":{"kind":"arxiv","id":"1309.3728","version":4},"attestation_state":"computed","paper":{"title":"Gaussian fluctuations for linear spectral statistics of large random covariance matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Jamal Najim, Jianfeng Yao","submitted_at":"2013-09-15T05:44:32Z","abstract_excerpt":"Consider a $N\\times n$ matrix $\\Sigma_n=\\frac{1}{\\sqrt{n}}R_n^{1/2}X_n$, where $R_n$ is a nonnegative definite Hermitian matrix and $X_n$ is a random matrix with i.i.d. real or complex standardized entries. The fluctuations of the linear statistics of the eigenvalues \\[\\operatorname {Trace}f \\bigl(\\Sigma_n\\Sigma_n^*\\bigr)=\\sum_{i=1}^Nf(\\lambda_i),\\qquad (\\lambda_i)\\ eigenvalues\\ of\\ \\Sigma_n\\Sigma_n^*,\\] are shown to be Gaussian, in the regime where both dimensions of matrix $\\Sigma_n$ go to infinity at the same pace and in the case where $f$ is of class $C^3$, that is, has three continuous de"},"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":"1309.3728","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-09-15T05:44:32Z","cross_cats_sorted":[],"title_canon_sha256":"3b2eacc8253356054788ce9c77a5a7f874c4508015a49da0e5d71eadbad54d47","abstract_canon_sha256":"bac0771e121a9478676a201efe25140322b4cd77e86f97f5d87188ec1e9ce9f1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:11:51.979948Z","signature_b64":"ZgNGlph4AzO2txMEq+SncyGauHnFEmYf6oNplPHIQb1FEOnXtRrI4sm/bjiN8J/C7GHDnBrIRJ9dOZdwOGUiCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1e19fb5f35d982fa5bca94a7d0063b72a7723458b51cbcd19a43f67deab30250","last_reissued_at":"2026-05-18T01:11:51.979574Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:11:51.979574Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Gaussian fluctuations for linear spectral statistics of large random covariance matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Jamal Najim, Jianfeng Yao","submitted_at":"2013-09-15T05:44:32Z","abstract_excerpt":"Consider a $N\\times n$ matrix $\\Sigma_n=\\frac{1}{\\sqrt{n}}R_n^{1/2}X_n$, where $R_n$ is a nonnegative definite Hermitian matrix and $X_n$ is a random matrix with i.i.d. real or complex standardized entries. The fluctuations of the linear statistics of the eigenvalues \\[\\operatorname {Trace}f \\bigl(\\Sigma_n\\Sigma_n^*\\bigr)=\\sum_{i=1}^Nf(\\lambda_i),\\qquad (\\lambda_i)\\ eigenvalues\\ of\\ \\Sigma_n\\Sigma_n^*,\\] are shown to be Gaussian, in the regime where both dimensions of matrix $\\Sigma_n$ go to infinity at the same pace and in the case where $f$ is of class $C^3$, that is, has three continuous de"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.3728","kind":"arxiv","version":4},"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":"1309.3728","created_at":"2026-05-18T01:11:51.979639+00:00"},{"alias_kind":"arxiv_version","alias_value":"1309.3728v4","created_at":"2026-05-18T01:11:51.979639+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.3728","created_at":"2026-05-18T01:11:51.979639+00:00"},{"alias_kind":"pith_short_12","alias_value":"DYM7WXZV3GBP","created_at":"2026-05-18T12:27:43.054852+00:00"},{"alias_kind":"pith_short_16","alias_value":"DYM7WXZV3GBPUW6K","created_at":"2026-05-18T12:27:43.054852+00:00"},{"alias_kind":"pith_short_8","alias_value":"DYM7WXZV","created_at":"2026-05-18T12:27:43.054852+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/DYM7WXZV3GBPUW6KSST5ABR3OK","json":"https://pith.science/pith/DYM7WXZV3GBPUW6KSST5ABR3OK.json","graph_json":"https://pith.science/api/pith-number/DYM7WXZV3GBPUW6KSST5ABR3OK/graph.json","events_json":"https://pith.science/api/pith-number/DYM7WXZV3GBPUW6KSST5ABR3OK/events.json","paper":"https://pith.science/paper/DYM7WXZV"},"agent_actions":{"view_html":"https://pith.science/pith/DYM7WXZV3GBPUW6KSST5ABR3OK","download_json":"https://pith.science/pith/DYM7WXZV3GBPUW6KSST5ABR3OK.json","view_paper":"https://pith.science/paper/DYM7WXZV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1309.3728&json=true","fetch_graph":"https://pith.science/api/pith-number/DYM7WXZV3GBPUW6KSST5ABR3OK/graph.json","fetch_events":"https://pith.science/api/pith-number/DYM7WXZV3GBPUW6KSST5ABR3OK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DYM7WXZV3GBPUW6KSST5ABR3OK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DYM7WXZV3GBPUW6KSST5ABR3OK/action/storage_attestation","attest_author":"https://pith.science/pith/DYM7WXZV3GBPUW6KSST5ABR3OK/action/author_attestation","sign_citation":"https://pith.science/pith/DYM7WXZV3GBPUW6KSST5ABR3OK/action/citation_signature","submit_replication":"https://pith.science/pith/DYM7WXZV3GBPUW6KSST5ABR3OK/action/replication_record"}},"created_at":"2026-05-18T01:11:51.979639+00:00","updated_at":"2026-05-18T01:11:51.979639+00:00"}