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Abstract <p> Let<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(X_1, X_2, \ldots , X_n\)</EquationSource> </InlineEquation> be independent random variables, and let<InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbf {E}^{(l)}\)</EquationSource> </InlineEquation> for<InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(l=1, \ldots , n\)</EquationSource> </InlineEquation> be symmetric deterministic matrices. We study matrices of the type<InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathbf {W} = \sum _{l=1}^n X_l \mathbf {E}^{(l)}\)</EquationSource> </InlineEquation>. The convergence of empirical spectral distribution of<InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\mathbf {W}\)</EquationSource> </InlineEquation> to the normal law is established under certain conditions on matrices<InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\mathbf {E}^{(l)}\)</EquationSource> </InlineEquation>. Applications include estimating the convergence rate for the spectraldistribution functions of palindromic and circulant random matrices. The analysis employs amethod introduced by the author in his 1980 work “On the Rate of Convergence in the CentralLimit Theorem for Weakly Dependent Variables”.</p>

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On Convergence of the Empirical Spectral Distribution Function for a Special Type of Random Matrices to the Normal Law

  • 期刊论文

  • 发表日期：

    2025年12月22日

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  • A. N. Tikhomirov

摘要

Abstract

Let \(X_1, X_2, \ldots , X_n\) be independent random variables, and let \(\mathbf {E}^{(l)}\) for \(l=1, \ldots , n\) be symmetric deterministic matrices. We study matrices of the type \(\mathbf {W} = \sum _{l=1}^n X_l \mathbf {E}^{(l)}\) . The convergence of empirical spectral distribution of \(\mathbf {W}\) to the normal law is established under certain conditions on matrices \(\mathbf {E}^{(l)}\) . Applications include estimating the convergence rate for the spectraldistribution functions of palindromic and circulant random matrices. The analysis employs amethod introduced by the author in his 1980 work “On the Rate of Convergence in the CentralLimit Theorem for Weakly Dependent Variables”.