  We establish asymptotic normality of weighted sums of periodograms of a
stationary linear process where weights depend on the sample size. Such sums
appear in numerous statistical applications and can be regarded as a
discretized versions of quadratic forms involving integrals of weighted
periodograms. Conditions for asymptotic normality of these weighted sums are
simple, minimal, and resemble Lindeberg-Feller condition for weighted sums of
independent and identically distributed random variables. Our results are
applicable to a large class of short, long or negative memory processes. The
proof is based on sharp bounds derived for Bartlett type approximation of these
sums by the corresponding sums of weighted periodograms of independent and
identically distributed random variables.
