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The random Hankel matrix converges to a sum of two self-adjoint variables each with symmetrized Rayleigh distribution."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The derivation rests on the structural connections between circulant, reverse-circulant and left skew-circulant matrices and the Toeplitz/Hankel matrices, together with the joint *-convergence of a random circulant matrix and a deterministic diagonal matrix."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"The authors establish *-convergence of random Toeplitz and Hankel matrices to sums of Gaussian or Rayleigh non-commutative variables via circulant connections, yielding new proofs and moment expressions for their limiting spectral distributions."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Random symmetric Toeplitz matrices converge in *-distribution to the sum of two non-commuting self-adjoint real Gaussian variables, while Hankel matrices converge to sums of symmetrized Rayleigh variables."}],"snapshot_sha256":"f0b3ad8e0f5fa2ecb4a0730f16e6cc0fa23a201330ae2166cbd0a63e1fceb86c"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[{"findings_count":0,"name":"doi_compliance","ran_at":"2026-05-19T19:01:30.908510Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"doi_title_agreement","ran_at":"2026-05-19T19:01:18.923992Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"cited_work_retraction","ran_at":"2026-05-19T18:22:01.893794Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"ai_meta_artifact","ran_at":"2026-05-19T17:33:31.037294Z","status":"skipped","version":"1.0.0"},{"findings_count":0,"name":"external_links","ran_at":"2026-05-19T17:31:48.018737Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"claim_evidence","ran_at":"2026-05-19T16:41:55.437638Z","status":"completed","version":"1.0.0"}],"endpoint":"/pith/2605.16160/integrity.json","findings":[],"snapshot_sha256":"24843f22d920c466e161b552d31e35f747df6d5446ef76132b19b903fc072a28","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We establish the joint $*$-convergence of a random circulant matrix and a specific deterministic diagonal matrix. We also show that the empirical spectral distributions of skew-circulant and left skew-circulant random matrices converge weakly a.s.~to complex Gaussian and symmetrized Rayleigh distributions, respectively.\n  The $*$-convergence of symmetric Toeplitz and Hankel random matrices is well known. So is the weak convergence of their random spectrum. However, not much is known about the limits. We exploit the connections of circulant, reverse circulant, and left skew-circulants with the ","authors_text":"Arup Bose, Pradeep Vishwakarma","cross_cats":[],"headline":"Random symmetric Toeplitz matrices converge in *-distribution to the sum of two non-commuting self-adjoint real Gaussian variables, while Hankel matrices converge to sums of symmetrized Rayleigh variables.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-05-15T16:39:43Z","title":"Revisiting Toeplitz and Hankel random matrices via $*$-convergence of circulant-type matrices"},"references":{"count":13,"internal_anchors":0,"resolved_work":13,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"Anderson, G.W., Guionnet, A. and Zeitouni, O. 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Probab.,17(82), 1-33","work_id":"ef9696f4-4b10-41d3-b967-c355127beec6","year":2012},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"Bose, A. and Saha, K. (2018).Random Circulant Matrices, Chapman & Hall","work_id":"820ebfc1-bf84-4eea-a48a-dbb52c6ce5d1","year":2018}],"snapshot_sha256":"5948e90dbf82cc372ea951bdc846f3f504b148db539451c2c2550a2e8faf45c0"},"source":{"id":"2605.16160","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-19T18:54:22.483242Z","id":"47cfd521-5639-430f-970c-c7361220ba73","model_set":{"reader":"grok-4.3"},"one_line_summary":"The authors establish *-convergence of random Toeplitz and Hankel matrices to sums of Gaussian or Rayleigh non-commutative variables via circulant connections, yielding new proofs and moment expressions for their limiting spectral distributions.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Random symmetric Toeplitz matrices converge in *-distribution to the sum of two non-commuting self-adjoint real Gaussian variables, while Hankel matrices converge to sums of symmetrized Rayleigh variables.","strongest_claim":"The random symmetric Toeplitz matrix converges in *-distribution to the sum of two non-commutative self-adjoint variables, each having a real Gaussian distribution. The random Hankel matrix converges to a sum of two self-adjoint variables each with symmetrized Rayleigh distribution.","weakest_assumption":"The derivation rests on the structural connections between circulant, reverse-circulant and left skew-circulant matrices and the Toeplitz/Hankel matrices, together with the joint *-convergence of a random circulant matrix and a deterministic diagonal matrix."}},"verdict_id":"47cfd521-5639-430f-970c-c7361220ba73"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:1c89959567c55168bebb069d5d5f4df6f3840a794d287331045c359f008c1646","target":"record","created_at":"2026-05-20T00:01:55Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"feeb69260a2b355d77939af877b3e1656cbeb5af825fe4ef5c35d87d165f28fa","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-05-15T16:39:43Z","title_canon_sha256":"3d0eb034e00835568dad97e4083b28390b7afc0d928df5f0997b63f6ba4ff650"},"schema_version":"1.0","source":{"id":"2605.16160","kind":"arxiv","version":1}},"canonical_sha256":"c9c57c7ae874ef22d4040c07a2f55cbb072e1c988b1f430b162653e20c721fb8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c9c57c7ae874ef22d4040c07a2f55cbb072e1c988b1f430b162653e20c721fb8","first_computed_at":"2026-05-20T00:01:55.515495Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T00:01:55.515495Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qMw+7xSFa8KV4dHoCbxVdLxZfBJvm7qascDMrr9rllui0q4hE6Ms/xRmU2CXqMSUt7est+qx2p8W16FsTkipAg==","signature_status":"signed_v1","signed_at":"2026-05-20T00:01:55.517307Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.16160","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1c89959567c55168bebb069d5d5f4df6f3840a794d287331045c359f008c1646","sha256:0dee0bebf9ec387cd637923c1ce71335cd61949022d204fc6911048ad9d514b1"],"state_sha256":"ba25704c0704ffe8d1f724b7eb26f1a3c382ccbd325608e9eb44baed751db729"}