Generalized Stable Multivariate Distribution and Anisotropic Dilations

Details Generalized Stable Multivariate Distribution and Anisotropic Dilations Generalized Stable Multivariate Distribution and Anisotropic Dilations

1999-12-09

arxiv.org/abs/chao-dyn/9912016v1

Nonlinear Sciences

Chaotic Dynamics

Scientific paper

After having closely re-examined the notion of a L\'evy's stable vector, it is shown that the notion of a stable multivariate distribution is more general than previously defined. Indeed, a more intrinsic vector definition is obtained with the help of non isotropic dilations and a related notion of generalized scale. In this framework, the components of a stable vector may not only have distinct Levy's stability indices $\alpha$'s, but the latter may depend on its norm. Indeed, we demonstrate that the Levy's stability index of a vector rather correspond to a linear application than to a scalar, and we show that the former should satisfy a simple spectral property.

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