21,248 research outputs found

    Asymptotic results for empirical measures of weighted sums of independent random variables

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    We prove that if a rectangular matrix with uniformly small entries and approximately orthogonal rows is applied to the independent standardized random variables with uniformly bounded third moments, then the empirical CDF of the resulting partial sums converges to the normal CDF with probability one. This implies almost sure convergence of empirical periodograms, almost sure convergence of spectra of circulant and reverse circulant matrices, and almost sure convergence of the CDF's generated from independent random variables by independent random orthogonal matrices. For special trigonometric matrices, the speed of the almost sure convergence is described by the normal approximation and by the large deviation principle

    A functional CLT for partial traces of random matrices

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    In this paper we show a functional central limit theorem for the sum of the first tn\lfloor t n \rfloor diagonal elements of f(Z)f(Z) as a function in tt, for ZZ a random real symmetric or complex Hermitian n×nn\times n matrix. The result holds for orthogonal or unitarily invariant distributions of ZZ, in the cases when the linear eigenvalue statistic trf(Z)\operatorname{tr} f(Z) satisfies a CLT. The limit process interpolates between the fluctuations of individual matrix elements as f(Z)1,1f(Z)_{1,1} and of the linear eigenvalue statistic. It can also be seen as a functional CLT for processes of randomly weighted measures
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