{"paper":{"title":"Extreme Eigenvalues of Large Dimensional Quaternion Sample Covariance Matrix","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Huiqin Li, Zhidong Bai","submitted_at":"2013-12-17T05:28:42Z","abstract_excerpt":"In this paper, we shall investigate the almost sure limits of the largest and smallest eigenvalues of a quaternion sample covariance matrix. Suppose that $\\mathbf X_n$ is a $p\\times n$ matrix whose elements are independent quaternion variables with mean zero, variance 1 and uniformly bounded fourth moments. Denote $\\mathbf S_n=\\frac{1}{n}\\mathbf X_n\\mathbf X_n^*$. In this paper, we shall show that $s_{\\max}\\left(\\mathbf S_n\\right)=s_{p}\\left(\\mathbf S_n\\right)\\to\\left(1+\\sqrt y\\right)^2, a.s.$ and $s_{\\min}\\left(\\mathbf S_n\\right)\\to\\left(1-\\sqrt y\\right)^2,a.s.$ as $n\\to\\infty$, where $y=\\lim"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.4649","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"}