Some Limit Theory for the Self-normalised Periodogram of Stable Processes

Abstract

Let X(t) = Sigma(j)(infinity) = (-infinity) psi(j)Z(t-j) be a discrete moving average process based on i.i.d. random variables (Z(t),)(t epsilon Z) with common distribution function from the domain of normal attraction of a p-stable law (0 <p less than or equal to 2). We prove weak convergence of the self-normalised periodogram [GRAPJICS] Furthermore, we show that smoothed versions of ($) over bar I-n,(X)(lambda) provide consistent estimates for the normalised transfer function for any p epsilon (0, 2] independent of p.