{"paper":{"title":"On asymptotic expansion and CLT of linear eigenvalue statistics for sample covariance matrices when $N/M\\rightarrow0$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Zhigang Bao","submitted_at":"2011-04-18T13:07:55Z","abstract_excerpt":"We study the renormalized real sample covariance matrix $H=X^TX/\\sqrt{MN}-\\sqrt{M/N}$ with $N/M\\rightarrow0$ as $N, M\\rightarrow \\infty$ in this paper. And we always assume $M=M(N)$. Here $X=[X_{jk}]_{M\\times N}$ is an $M\\times N$ real random matrix with i.i.d entries, and we assume $\\mathbb{E}|X_{11}|^{5+\\delta}<\\infty$ with some small positive $\\delta$. The Stieltjes transform $m_N(z)=N^{-1}Tr(H-z)^{-1}$ and the linear eigenvalue statistics of $H$ are considered. We mainly focus on the asymptotic expansion of $\\mathbb{E}\\{m_N(z)\\}$ in this paper. Then for some fine test function, a central l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.3470","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"}