{"paper":{"title":"The Central Limit Theorem for Linear Eigenvalue Statistics of the Sum of Independent Matrices of Rank One","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"A. Lytova, A. Pajor, L. Pastur, O. Gu\\'edon","submitted_at":"2013-10-09T14:46:59Z","abstract_excerpt":"We consider $n\\times n$ random matrices $M_{n}=\\sum_{\\alpha =1}^{m}{\\tau _{\\alpha }}\\mathbf{y}_{\\alpha }\\otimes \\mathbf{y}_{\\alpha }$, where $\\tau _{\\alpha }\\in \\mathbb{R}$, $\\{\\mathbf{y}_{\\alpha }\\}_{\\alpha =1}^{m}$ are i.i.d. isotropic random vectors of $\\mathbb{R}^n$, whose components are not necessarily independent. It was shown in arXiv:0710.1346 that if $m,n\\rightarrow \\infty$, $m/n\\rightarrow c\\in \\lbrack 0,\\infty )$, the Normalized Counting Measures of $\\{\\tau _{\\alpha }\\}_{\\alpha =1}^{m}$ converge weakly and $\\{\\mathbf{y}_\\alpha\\}_{\\alpha=1}^m$ are \\textit{good} (see corresponding def"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.2506","kind":"arxiv","version":2},"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"}