{"paper":{"title":"Convergence of the largest eigenvalue of normalized sample covariance matrices when p and n both tend to infinity with their ratio converging to zero","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"B. B. Chen, G. M. Pan","submitted_at":"2012-11-23T12:13:41Z","abstract_excerpt":"Let $\\mathbf{X}_p=(\\mathbf{s}_1,...,\\mathbf{s}_n)=(X_{ij})_{p \\times n}$ where $X_{ij}$'s are independent and identically distributed (i.i.d.) random variables with $EX_{11}=0,EX_{11}^2=1$ and $EX_{11}^4<\\infty$. It is showed that the largest eigenvalue of the random matrix $\\mathbf{A}_p=\\frac{1}{2\\sqrt{np}}(\\mathbf{X}_p\\mathbf{X}_p^{\\prime}-n\\mathbf{I}_p)$ tends to 1 almost surely as $p\\rightarrow\\infty,n\\rightarrow\\infty$ with $p/n\\rightarrow0$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.5479","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"}