{"paper":{"title":"Convergence of the empirical spectral distribution function of Beta matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Guangming Pan, Jiang Hu, Wang Zhou, Zhidong Bai","submitted_at":"2012-08-29T16:04:47Z","abstract_excerpt":"Let $\\mathbf{B}_n=\\mathbf {S}_n(\\mathbf {S}_n+\\alpha_n\\mathbf {T}_N)^{-1}$, where $\\mathbf {S}_n$ and $\\mathbf {T}_N$ are two independent sample covariance matrices with dimension $p$ and sample sizes $n$ and $N$, respectively. This is the so-called Beta matrix. In this paper, we focus on the limiting spectral distribution function and the central limit theorem of linear spectral statistics of $\\mathbf {B}_n$. Especially, we do not require $\\mathbf {S}_n$ or $\\mathbf {T}_N$ to be invertible. Namely, we can deal with the case where $p>\\max\\{n,N\\}$ and $p<n+N$. Therefore, our results cover many "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.5953","kind":"arxiv","version":3},"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"}