Spectral Density Estimate for Stable Processes Observed with an Additive Error

In this paper, a symmetric alpha stable process where its spectral representation has an additive error is considered. The error is supposed to be constant. A periodograrn as estimator of the spectral density and its rate of convergence are given. In order to give an asymptotically unbiased and consistent estimate of the spectral density, this periodogram is smoothed by an adapted spectral window. The rate of convergence is given.