Götze, F.; Tikhomirov, A. Rate of convergence to the semi-circular law. (English) Zbl 1031.60019 Probab. Theory Relat. Fields 127, No. 2, 228-276 (2003). Let \(X_{jk}\), \(1\leq j\leq k < \infty\), be independent random variables with \(EX_{jk}=0\) and \(EX_{jk}^2=1\) and let \(X_{kj}=X_{jk}\) for \(1\leq j<k< \infty\). For a fixed \(n\geq 1,\) denote by \(\lambda_1 \leq\ldots\leq \lambda_n\) the eigenvalues of the symmetric \(n\times n\) matrix \(W= (W_{jk})_{j,k=1}^n\), \( W_{jk}=X_{jk}\) for \(1\leq j\leq k\leq n\), and define their spectral distribution function \(F_n(x)=\frac{1}{n}\sum_{j=1}^n I_{\{\lambda_j \leq x\sqrt{n} \}}\), where \(I_{\{B\}}\) denotes the indicator of an event \(B\). Let \(g(x)\) and \(G(x)\) denote the density and the distribution function of the standard semi-circular law, respectively, that is \[ g(x)=\frac{1}{2\pi}\sqrt{4-x^2}{I}_{\{|x|\leq 2 \}},\quad G(x)=\int_{-\infty}^x g(u) du. \] Set \[ \Delta_n = \sup_x |{E}F_n(x) - G(x)|,\qquad \Delta_n^{*} ={E} \sup_x |F_n(x) - G(x)|. \] The main results are given by the following: Theorem 1.1. Assume that \(X_{jk}\) for \(k\geq 1\) satisfy the conditions above and that \(M_4:= M_4^{(n)}:=\sup_{1\leq j\leq k\leq n} {\mathbf E}|X_{jk}|^4 <\infty\). Then there exsists an absolute constant \(C> 0\) such that, for any \(n\geq 1,\) \[ \Delta_n = \sup_x |{\mathbf E}F_n(x) - G(x)|\leq CM_{4}^{1/2} n^{-1/2}. \] Theorem 1.2. Assume that \(X_{jk}\) for \(j,k\geq 1\) satisfy the conditions above and that \(M_{12}:= M_{12}^{(n)}:=\sup_{1\leq j\leq k\leq n} {\mathbf E}|X_{jk}|^{12} <\infty\). Then there exsists an absolute constant \(C> 0\) such that, for any \(n\geq 1,\) \[ \Delta_n^{*} ={\mathbf E} \sup_x |F_n(x) - G(x)|\leq CM_{12}^{1/6} n^{-1/2}. \] Reviewer: Zdzisław Rychlik (Lublin) Cited in 1 ReviewCited in 35 Documents MSC: 60F05 Central limit and other weak theorems 60G50 Sums of independent random variables; random walks Keywords:independent random variables; spectral distribution; random matrix PDFBibTeX XMLCite \textit{F. Götze} and \textit{A. Tikhomirov}, Probab. Theory Relat. Fields 127, No. 2, 228--276 (2003; Zbl 1031.60019) Full Text: DOI arXiv