Properties of spectral covariance for linear processes with infinite variance

Abstract

We consider a measure of dependence for symmetric α-stable random vectors, which was introduced by the second author in 1976. We demonstrate that this measure of dependence, which we suggest to call the spectral covariance, can be extended to random vectors in the domain of normal attraction of general stable vectors. We investigate the asymptotic of the spectral covariance function for linear stable (Ornstein–Uhlenbeck, log-fractional, linear-fractional) processes with infinite variance and show that, in comparison with the results on the properties of codifference of these processes, obtained two decades ago, the results for the spectral variance are obtained under more general conditions and calculations are simpler.