Limit theorems for stable processes with application to spectral density estimation

This paper deals with issues pertaining to estimating the spectral density of a stationary harmonizable α-stable process, where 0 < α < 2. The estimator we consider is obtained by smoothing the periodogram, which has a similar flavor as the usual kernel spectral density estimator for a second-order stationary process. We derive the basic asymptotic properties of the estimator and show how to pick the optimal smoothing parameter for α in different intervals of (0, 2). At the heart of these derivations is the theoretical problem of finding the asymptotic distribution of a weighted average of |Y(u)|p over an increasing interval, where 0 < p < ∞ and Y is a nearly stationary moving average α-stable process. Our results partially extend the limit theorems in Davis (1983) and LePage et al. (1981).