A LIMIT THEOREM FOR QUADRATIC FORMS AND ITS APPLICATIONS

We consider quadratic forms of martingale differences and establish a central limit theorem under mild and easily verifiable conditions+ By approximating Fourier transforms of stationary processes by martingales, our central limit theorem is applied to the smoothed periodogram estimate of spectral density functions+ Our results go beyond earlier ones by allowing a variety of nonlinear time series and by avoiding strong mixing and0or summability conditions on joint cumulants+