Generalized Stable Multivariate Distribution and Anisotropic Dilations

TL;DR

This paper introduces a more general framework for multivariate stable distributions using anisotropic dilations, allowing components to have different stability indices and a vector-based stability measure.

Contribution

It provides a new intrinsic vector-based definition of stable multivariate distributions incorporating non-isotropic dilations and a spectral property for stability indices.

Findings

Stable vectors can have stability indices depending on their norm.

The stability index corresponds to a linear application, not a scalar.

A simple spectral property characterizes the stability indices.

Abstract

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.