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We introduce the notion of symmetric covariation, which is a new measure ofdependence between two components of a symmetric $\alpha$-stable random vector,where the stability parameter $\alpha$ measures the heavy-tailedness of itsdistribution. Unlike covariation that exists only when $\alpha\in(1,2]$,symmetric covariation is well defined for all $\alpha\in(0,2]$. We show thatsymmetric covariation can be defined using the proposed generalized fractionalderivative, which has broader usages than those involved in this work. Severalproperties of symmetric covariation have been derived. These are either similarto or more general than those of the covariance functions in the Gaussian case.The main contribution of this framework is the representation of thecharacteristic function of bivariate symmetric $\alpha$-stable distribution viaconvergent series based on a sequence of symmetric covariations. This seriesrepresentation extends the one of bivariate Gaussian.

Series Representation of Jointly S$\alpha$S Distribution via Symmetric Covariations