We consider random matrix ensembles on the Hermitian matrices that are heavy tailed, in particular not all moments exist, and that are invariant under the conjugate action of the unitary group. The latter property entails that the eigenvectors are Haar distributed and, therefore, factorise from the eigenvalue statistics. We prove a classification for stable matrix ensembles of this kind of matrices represented in terms of matrices, their eigenvalues and their diagonal entries with the help of the classification of the multivariate stable distributions and the harmonic analysis on symmetric matrix spaces. Moreover, we identify sufficient and necessary conditions for their domains of attraction. To illustrate our findings we discuss for instance elliptical invariant random matrix ensembles and P\'olya ensembles. As a byproduct we generalise the derivative principle on the Hermitian matrices to general tempered distributions.
翻译:我们认为,在埃米提亚矩阵上随机的矩阵集合是大量尾随的,特别是并不是所有时刻都存在,而且根据单一组群的共鸣动作是变化无常的。后一种属性意味着向生者是Haar分布的,因此,从egenvaly统计数据中考虑到。我们证明,在对多变量稳定分布的分类和对对称矩阵空间的调和分析的帮助下,对以矩阵、其电子元值及其对角条目为代表的这类矩阵组合的稳定矩阵组合进行了分类。此外,我们为它们的吸引力领域确定了足够和必要的条件。为了说明我们讨论的结果,例如,椭丽花异随机矩阵组合和P\'olya酶组合。作为副产品,我们将赫米提亚矩阵的衍生原则推广到一般的温和分布中。