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We derive a central limit theorem for the ordered singular spectrum $\\sigma_{sing}(S_k)$ with a normal distribution as limit with explicit analytic formulas for the drift vector and the covariance matrix.\n  The main ingredient for the proof will be a oscillatory result for the spherical functions $\\phi_{i\\rho+\\lambda}$ of $(GL_n(\\b F),U_n(\\b F))$. More precisely, we present a necessarily uni"},"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":"1205.4866","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-05-22T10:12:42Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"6220d30ea911cb9a8a02de2dacece367e370f56515d29b3eff1c4ddd0d1533cf","abstract_canon_sha256":"8b107a835f27362b56893732d748a51a1982cd68e13979b3d62c47fde2318a0c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:55:11.473249Z","signature_b64":"hVzfE41v6mB8rRLcPLFJrvpi9DhBDUKJNuLwlc75ipzxUems0T78Vg0Ix8ZfHqrXpF4klMNrvn1t8a31bX1/Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"14f5f976bd5efb7dcc186fe503c9f9a8edfc5ca3947b96a1a5168779a8a8aecd","last_reissued_at":"2026-05-18T03:55:11.472620Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:55:11.472620Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Uniform oscillatory behavior of spherical functions of $GL_n/U_n$ at the identity and a central limit theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CA","authors_text":"Michael Voit","submitted_at":"2012-05-22T10:12:42Z","abstract_excerpt":"Let $\\mathbb F=\\mathbb R$ or $\\mathbb C$ and $n\\in\\b N$.\n  Let $(S_k)_{k\\ge0}$ be a time-homogeneous random walk on $GL_n(\\b F)$ associated with an $U_n(\\b F)$-biinvariant measure $\\nu\\in M^1(GL_n(\\b F))$. We derive a central limit theorem for the ordered singular spectrum $\\sigma_{sing}(S_k)$ with a normal distribution as limit with explicit analytic formulas for the drift vector and the covariance matrix.\n  The main ingredient for the proof will be a oscillatory result for the spherical functions $\\phi_{i\\rho+\\lambda}$ of $(GL_n(\\b F),U_n(\\b F))$. More precisely, we present a necessarily uni"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.4866","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":"1205.4866","created_at":"2026-05-18T03:55:11.472723+00:00"},{"alias_kind":"arxiv_version","alias_value":"1205.4866v1","created_at":"2026-05-18T03:55:11.472723+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.4866","created_at":"2026-05-18T03:55:11.472723+00:00"},{"alias_kind":"pith_short_12","alias_value":"CT27S5V5L35X","created_at":"2026-05-18T12:27:01.376967+00:00"},{"alias_kind":"pith_short_16","alias_value":"CT27S5V5L35X3TAY","created_at":"2026-05-18T12:27:01.376967+00:00"},{"alias_kind":"pith_short_8","alias_value":"CT27S5V5","created_at":"2026-05-18T12:27:01.376967+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/CT27S5V5L35X3TAYN7SQHSPZVD","json":"https://pith.science/pith/CT27S5V5L35X3TAYN7SQHSPZVD.json","graph_json":"https://pith.science/api/pith-number/CT27S5V5L35X3TAYN7SQHSPZVD/graph.json","events_json":"https://pith.science/api/pith-number/CT27S5V5L35X3TAYN7SQHSPZVD/events.json","paper":"https://pith.science/paper/CT27S5V5"},"agent_actions":{"view_html":"https://pith.science/pith/CT27S5V5L35X3TAYN7SQHSPZVD","download_json":"https://pith.science/pith/CT27S5V5L35X3TAYN7SQHSPZVD.json","view_paper":"https://pith.science/paper/CT27S5V5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1205.4866&json=true","fetch_graph":"https://pith.science/api/pith-number/CT27S5V5L35X3TAYN7SQHSPZVD/graph.json","fetch_events":"https://pith.science/api/pith-number/CT27S5V5L35X3TAYN7SQHSPZVD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CT27S5V5L35X3TAYN7SQHSPZVD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CT27S5V5L35X3TAYN7SQHSPZVD/action/storage_attestation","attest_author":"https://pith.science/pith/CT27S5V5L35X3TAYN7SQHSPZVD/action/author_attestation","sign_citation":"https://pith.science/pith/CT27S5V5L35X3TAYN7SQHSPZVD/action/citation_signature","submit_replication":"https://pith.science/pith/CT27S5V5L35X3TAYN7SQHSPZVD/action/replication_record"}},"created_at":"2026-05-18T03:55:11.472723+00:00","updated_at":"2026-05-18T03:55:11.472723+00:00"}