{"paper":{"title":"On spectrum of sample correlation matrices from large fold tensor vectors","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Sample correlation matrices from k-fold tensor vectors converge to the Marčenko-Pastur law when k grows slower than n.","cross_cats":[],"primary_cat":"math.PR","authors_text":"Wangjun Yuan","submitted_at":"2026-04-08T08:40:28Z","abstract_excerpt":"In this paper, we investigate the limiting spectral distribution of the sample correlation matrix, whose sample vectors are $k$-fold tensor products of $n$-dimensional vectors with i.i.d. entries. We focus on the limiting regime $n,k \\to \\infty$ with $k = o(n)$, and we show that the limiting spectral distribution is the Mar\\v{c}enko-Pastur law. As a consequence, we show that the limiting spectral distribution of the Whishart matrix from the $k$-fold tensor product of independent uniformly distributed unit vectors in $\\mathbb C^n$ is the Mar\\v{c}enko-Pastur law."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we show that the limiting spectral distribution is the Marčenko-Pastur law. As a consequence, we show that the limiting spectral distribution of the Wishart matrix from the k-fold tensor product of independent uniformly distributed unit vectors in C^n is the Marčenko-Pastur law.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The regime n, k → ∞ with k = o(n) together with the assumption that the base vectors have i.i.d. entries (or are uniformly distributed on the unit sphere) is sufficient for the tensor-structured sample matrices to inherit the Marchenko-Pastur limiting law.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The limiting spectral distribution of sample correlation matrices from k-fold tensor vectors is the Marčenko-Pastur law when n, k → ∞ with k = o(n).","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Sample correlation matrices from k-fold tensor vectors converge to the Marčenko-Pastur law when k grows slower than n.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"a05f72a812d2d6d9ddd0caa26d78c085818bf6a323b2de4333d22449777200c3"},"source":{"id":"2604.06823","kind":"arxiv","version":2},"verdict":{"id":"aad75aa9-4f81-4dcf-8b6c-1060dde5931b","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T18:18:45.655757Z","strongest_claim":"we show that the limiting spectral distribution is the Marčenko-Pastur law. As a consequence, we show that the limiting spectral distribution of the Wishart matrix from the k-fold tensor product of independent uniformly distributed unit vectors in C^n is the Marčenko-Pastur law.","one_line_summary":"The limiting spectral distribution of sample correlation matrices from k-fold tensor vectors is the Marčenko-Pastur law when n, k → ∞ with k = o(n).","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The regime n, k → ∞ with k = o(n) together with the assumption that the base vectors have i.i.d. entries (or are uniformly distributed on the unit sphere) is sufficient for the tensor-structured sample matrices to inherit the Marchenko-Pastur limiting law.","pith_extraction_headline":"Sample correlation matrices from k-fold tensor vectors converge to the Marčenko-Pastur law when k grows slower than n."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.06823/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"d7d76f1032ecb17208ea9e7680914c06b6ee257e5110ec938808b3658f429821"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}