Nonlinear Functionals of the Periodogram

A central limit theorem is stated for a wide class of triangular arrays of nonlinear functionals of the periodogram of a stationary linear sequence. Those functionals may be singular and not-bounded. The proof of this result is based on Bartlett decomposition and an existing counterpart result for the periodogram of an independent and identically distributed sequence, here taken to be the driving noise. The main contribution of this paper is to prove the asymptotic negligibility of the remainder term from Bartlett decomposition, feasible under short dependence assumption. As it is highlighted by applications (to estimation of nonlinear functionals of the spectral density, robust spectral estimation, local polynomial approximation and log-periodogram regression), this extends may results until then tied to Gaussian assumption