{"paper":{"title":"Fluctuation of Eigenvalues for Random Toeplitz and Related Matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Dang-Zheng Liu, Xin Sun, Zheng-Dong Wang","submitted_at":"2010-10-17T04:04:29Z","abstract_excerpt":"Consider random symmetric Toeplitz matrices $T_{n}=(a_{i-j})_{i,j=1}^{n}$ with matrix entries $a_{j}, j=0,1,2,...,$ being independent real random variables such that\n  \\be \\mathbb{E}[a_{j}]=0, \\ \\ \\mathbb{E}[|a_{j}|^{2}]=1 \\ \\ \\textrm{for}\\,\\ \\ j=0,1,2,...,\\ee (homogeneity of 4-th moments) \\be{\\kappa=\\mathbb{E}[|a_{j}|^{4}],}\\ee\n  \\noindent and further (uniform boundedness)\\be\\sup\\limits_{j\\geq 0} \\mathbb{E}[|a_{j}|^{k}]=C_{k}<\\iy\\ \\ \\ \\textrm{for} \\ \\ \\ k\\geq 3.\\ee\n  Under the assumption of $a_{0}\\equiv 0$, we prove a central limit theorem for linear statistics of eigenvalues for a fixed poly"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.3394","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"}