{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:AG7ZY6TOZ2RQ3Y3D7BMXPRHF63","short_pith_number":"pith:AG7ZY6TO","schema_version":"1.0","canonical_sha256":"01bf9c7a6ecea30de363f85977c4e5f6d7cbe8befd5fa18826cefbfae581b96c","source":{"kind":"arxiv","id":"1104.3470","version":3},"attestation_state":"computed","paper":{"title":"On asymptotic expansion and CLT of linear eigenvalue statistics for sample covariance matrices when $N/M\\rightarrow0$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Zhigang Bao","submitted_at":"2011-04-18T13:07:55Z","abstract_excerpt":"We study the renormalized real sample covariance matrix $H=X^TX/\\sqrt{MN}-\\sqrt{M/N}$ with $N/M\\rightarrow0$ as $N, M\\rightarrow \\infty$ in this paper. And we always assume $M=M(N)$. Here $X=[X_{jk}]_{M\\times N}$ is an $M\\times N$ real random matrix with i.i.d entries, and we assume $\\mathbb{E}|X_{11}|^{5+\\delta}<\\infty$ with some small positive $\\delta$. The Stieltjes transform $m_N(z)=N^{-1}Tr(H-z)^{-1}$ and the linear eigenvalue statistics of $H$ are considered. We mainly focus on the asymptotic expansion of $\\mathbb{E}\\{m_N(z)\\}$ in this paper. Then for some fine test function, a central l"},"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":"1104.3470","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-04-18T13:07:55Z","cross_cats_sorted":[],"title_canon_sha256":"eda3c81e9b1872d84e33f86ea577f647d379d7254a464e3f79d878e168582588","abstract_canon_sha256":"60720b355a19065a2e219d78f4dea1145dc554500fa066bcb508d60680f49db3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:08:22.225146Z","signature_b64":"G+jyA38C+pnDPWY7fzSg5AkVzrCrjcRhc5iL8XwXoqnCbUKWFZtSx3BMWsYrsilW3+t6qhJY15G9YGMvtAWABA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"01bf9c7a6ecea30de363f85977c4e5f6d7cbe8befd5fa18826cefbfae581b96c","last_reissued_at":"2026-05-18T04:08:22.224664Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:08:22.224664Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On asymptotic expansion and CLT of linear eigenvalue statistics for sample covariance matrices when $N/M\\rightarrow0$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Zhigang Bao","submitted_at":"2011-04-18T13:07:55Z","abstract_excerpt":"We study the renormalized real sample covariance matrix $H=X^TX/\\sqrt{MN}-\\sqrt{M/N}$ with $N/M\\rightarrow0$ as $N, M\\rightarrow \\infty$ in this paper. And we always assume $M=M(N)$. Here $X=[X_{jk}]_{M\\times N}$ is an $M\\times N$ real random matrix with i.i.d entries, and we assume $\\mathbb{E}|X_{11}|^{5+\\delta}<\\infty$ with some small positive $\\delta$. The Stieltjes transform $m_N(z)=N^{-1}Tr(H-z)^{-1}$ and the linear eigenvalue statistics of $H$ are considered. We mainly focus on the asymptotic expansion of $\\mathbb{E}\\{m_N(z)\\}$ in this paper. Then for some fine test function, a central l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.3470","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":"1104.3470","created_at":"2026-05-18T04:08:22.224738+00:00"},{"alias_kind":"arxiv_version","alias_value":"1104.3470v3","created_at":"2026-05-18T04:08:22.224738+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.3470","created_at":"2026-05-18T04:08:22.224738+00:00"},{"alias_kind":"pith_short_12","alias_value":"AG7ZY6TOZ2RQ","created_at":"2026-05-18T12:26:24.575870+00:00"},{"alias_kind":"pith_short_16","alias_value":"AG7ZY6TOZ2RQ3Y3D","created_at":"2026-05-18T12:26:24.575870+00:00"},{"alias_kind":"pith_short_8","alias_value":"AG7ZY6TO","created_at":"2026-05-18T12:26:24.575870+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/AG7ZY6TOZ2RQ3Y3D7BMXPRHF63","json":"https://pith.science/pith/AG7ZY6TOZ2RQ3Y3D7BMXPRHF63.json","graph_json":"https://pith.science/api/pith-number/AG7ZY6TOZ2RQ3Y3D7BMXPRHF63/graph.json","events_json":"https://pith.science/api/pith-number/AG7ZY6TOZ2RQ3Y3D7BMXPRHF63/events.json","paper":"https://pith.science/paper/AG7ZY6TO"},"agent_actions":{"view_html":"https://pith.science/pith/AG7ZY6TOZ2RQ3Y3D7BMXPRHF63","download_json":"https://pith.science/pith/AG7ZY6TOZ2RQ3Y3D7BMXPRHF63.json","view_paper":"https://pith.science/paper/AG7ZY6TO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1104.3470&json=true","fetch_graph":"https://pith.science/api/pith-number/AG7ZY6TOZ2RQ3Y3D7BMXPRHF63/graph.json","fetch_events":"https://pith.science/api/pith-number/AG7ZY6TOZ2RQ3Y3D7BMXPRHF63/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AG7ZY6TOZ2RQ3Y3D7BMXPRHF63/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AG7ZY6TOZ2RQ3Y3D7BMXPRHF63/action/storage_attestation","attest_author":"https://pith.science/pith/AG7ZY6TOZ2RQ3Y3D7BMXPRHF63/action/author_attestation","sign_citation":"https://pith.science/pith/AG7ZY6TOZ2RQ3Y3D7BMXPRHF63/action/citation_signature","submit_replication":"https://pith.science/pith/AG7ZY6TOZ2RQ3Y3D7BMXPRHF63/action/replication_record"}},"created_at":"2026-05-18T04:08:22.224738+00:00","updated_at":"2026-05-18T04:08:22.224738+00:00"}