Spectral estimates and stable processes

AbstractLet Xt = ∑j = −∞∞ ψjZt−j be a discrete time moving average process based on i.i.d. symmetric random variables {Zt} with a common distribution function from the domain of normal attraction of a p-stable law (0 < p < 2). We derive the limit distribution of the normalized periodogram In,X(λ) = |n−1p ∑t = 1n Xt e−itλ|2, −π ⩽ λ ⩽ π. This generalizes the classical result for p = 2. In contrast to the classical case, for values 0 < λ1 < ⋯ < λm < π the periodogram ordinates In, X(λi), i = 1, …, m, are not asymptotically independent