  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.
