SPECTRAL ESTIMATES AND STABLE PROCESSES

Let X(t) = SIGMA(j=-infinity)infinity psi(j)Z(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(lambda) = \n-1/p SIGMA(t=1)n X(t) e(-itlambda)\2, -pi less-than-or-equal-to lambda less-than-or-equal-to pi. This generalizes the classical result for p = 2. In contrast to the classical case, for values 0 < lambda1 < ... < lambda(m) < pi the periodogram ordinates I(n),X(lambda(i)), i = 1,..., m, are not asymptotically independent.