Spectral estimates and stable processes

Let X_t = ∑ j = -∞ ∞ ψ_jZ_t-j be a discrete time moving average process based on i.i.d. symmetric random variables {Z_t} 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 I_n,X(λ) = |n ^{-1 p} ∑ t = 1 n X_t e^-itλ|², -π ≤ λ ≤ π. This generalizes the classical result for p = 2. In contrast to the classical case, for values 0 < λ₁ < ⋯ < λ_m < π the periodogram ordinates I_{n, X}(λ_i), i = 1, ..., m, are not asymptotically independent.