On $\alpha$-covariance, long, short and negative memories for sequences of random variables with infinite variance

Abstract

We consider a measure of dependence for symmetric $\alpha$-stable random vectors, which was introduced by the author in 1976. We demonstrate that this measure of dependence can be extended for much more broad class of random vectors (up to regularly varying vectors in separable Banach spaces). This measure is applied for linear random processes and fields with heavy-tailed innovations, for some stable processes, and these applications show that this dependence measure, named as $\alpha$-covariance is a good substitute for the usual covariance. Also we discuss a problem of defining long, short, and negative memories for stationary processes and fields with infinite variances.