Some asymptotic theory for sums of weighted periodograms 1

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 the quadratic forms involving integrals of weighted periodograms. Conditions for asymptotic nor-mality of these weighted sums are simple and resemble Lindeberg-Feller condition for weighted sums of independent and identically distributed random variables. Our re-sults are valid for short, long or negative memory processes. The proof is based on sharp bounds derived for Bartlett type approximation of these sums by the correspond-ing sums of weighted periodograms of independent and identically distributed random variables.