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Rate of convergence to the semi-circular law. (English) Zbl 1031.60019

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 \(n\geq 1,\) \[ \Delta_n^{*} ={\mathbf E} \sup_x |F_n(x) - G(x)|\leq CM_{12}^{1/6} n^{-1/2}. \]

MSC:

60F05 Central limit and other weak theorems
60G50 Sums of independent random variables; random walks
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