{"paper":{"title":"Gaussian fluctuations for linear spectral statistics of large random covariance matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Jamal Najim, Jianfeng Yao","submitted_at":"2013-09-15T05:44:32Z","abstract_excerpt":"Consider a $N\\times n$ matrix $\\Sigma_n=\\frac{1}{\\sqrt{n}}R_n^{1/2}X_n$, where $R_n$ is a nonnegative definite Hermitian matrix and $X_n$ is a random matrix with i.i.d. real or complex standardized entries. The fluctuations of the linear statistics of the eigenvalues \\[\\operatorname {Trace}f \\bigl(\\Sigma_n\\Sigma_n^*\\bigr)=\\sum_{i=1}^Nf(\\lambda_i),\\qquad (\\lambda_i)\\ eigenvalues\\ of\\ \\Sigma_n\\Sigma_n^*,\\] are shown to be Gaussian, in the regime where both dimensions of matrix $\\Sigma_n$ go to infinity at the same pace and in the case where $f$ is of class $C^3$, that is, has three continuous de"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.3728","kind":"arxiv","version":4},"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"}