We introduce the notion of symmetric covariation, which is a new measure of dependence between two components of a symmetric \(\alpha \)-stable random vector, where the stability parameter \(\alpha \) measures the heavy-tailedness of its distribution. Unlike covariation that exists only when \(\alpha \in (1,2]\), symmetric covariation is well defined for all \(\alpha \in (0,2]\). We show that symmetric covariation can be defined using the proposed generalized fractional derivative, which has broader usages than those involved in this work. Several properties of symmetric covariation have been derived. These are either similar to or more general than those of the covariance functions in the Gaussian case. The main contribution of this framework is the representation of the characteristic function of bivariate symmetric \(\alpha \)-stable distribution via convergent series based on a sequence of symmetric covariations. This series representation extends the one of bivariate Gaussian.

我们介绍对称协变的概念，这是对称\（\ alpha \）稳定随机向量的两个分量之间依存关系的新度量，其中稳定性参数\（\ alpha \）衡量其分布的重尾性。与仅当\（\ alpha \ in（1,2] \）时才存在的协变不同，对称的协变可以很好地定义所有\（\ alpha \ in（0,2] \）。我们表明，可以使用提出的广义分数导数来定义对称协方差，该广义分数导数的使用范围比本工作所涉及的广泛。对称协变的几种性质已经得到了推导。这些与高斯情况下的协方差函数相似，或更笼统。该框架的主要贡献是通过基于对称协变序列的收敛级数来表示双变量对称\（\ alpha \） -稳定分布的特征函数。该级数表示扩展了二元高斯型之一。