CENTRAL LIMIT THEOREM FOR LINEAR EIGENVALUE STATISTICS OF RANDOM MATRICES WITH INDEPENDENT ENTRIES

• 成果类型: Article
• 署名作者: Lytova, A.; Pastur, L.
• 署名单位: National Academy of Sciences Ukraine; B. Verkin Institute for Low Temperature Physics & Engineering of the National Academy of Sciences of Ukraine
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/09-AOP452
• 发表日期: 2009
• 页码: 1778-1840
• 关键词: fluctuations; functionals; clt

摘要

We consider n x n real symmetric and Hermitian Wigner random matrices n(-1/2)W with independent (modulo symmetry condition) entries and the (null) sample covariance matrices n(-1) X*X with independent entries of m x n matrix X. Assuming first that the 4th cumulant (excess) kappa(4) of entries of W and X is zero and that their 4th moments satisfy a Lindeberg type condition, we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem (CLT) as n -> infinity, m -> infinity, m/n -> C is an element of [0, infinity) with the same variance as for Gaussian matrices if the test functions of statistics are smooth enough (essentially of the class C-5). This is done by using a simple interpolation trick from the known results for the Gaussian matrices and the integration by parts, presented in the form of certain differentiation formulas. Then, by using a more elaborated version of the techniques, we prove the CLT in the case of nonzero excess of entries again for essentially C-5 test function. Here the variance of statistics contains an additional term proportional to kappa(4). The proofs of all limit theorems follow essentially the same scheme.