Asymptotic results for empirical measures of weighted sums of independent random variables
Bernard Bercu, Universite Bordeaux 1
Wlodek Bryc, University of Cincinnati
Abstract
We investigate the asymptotic behavior of weighted sums of
independent standardized random variables with uniformly bounded
third moments. The sequence of weights is given by a family of rectangular matrices
with uniformly small entries and approximately orthogonal rows.
We prove that the empirical CDF of the resulting partial sums converges to the normal
CDF with probability one.
This result implies almost sure convergence of empirical periodograms, almost sure
convergence of spectral distribution of circulant and reverse circulant matrices,
and almost sure convergence of the CDF generated from independent
random variables by independent random orthogonal matrices.
In the special case of trigonometric weights, the speed of the almost sure
convergence is described by a normal approximation as well as a
large deviation principle.
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