21,248 research outputs found
Asymptotic results for empirical measures of weighted sums of independent random variables
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
In this paper we show a functional central limit theorem for the sum of the
first diagonal elements of as a function in ,
for a random real symmetric or complex Hermitian matrix. The
result holds for orthogonal or unitarily invariant distributions of , in the
cases when the linear eigenvalue statistic satisfies a
CLT. The limit process interpolates between the fluctuations of individual
matrix elements as 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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