We consider random matrix ensembles on the set of 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, the latter playing a particular role in matrix convolutions. As a byproduct we generalise the derivative principle on the Hermitian matrices to general tempered distributions. This principle relates the joint probability density of the eigenvalues and the diagonal entries of the random matrix.

翻译：我们研究埃尔米特矩阵集合上的重尾随机矩阵系综，特别是存在矩缺失的情形，且这些系综在酉群的共轭作用下保持不变。后一性质意味着特征向量服从哈尔分布，因此可从特征值统计中分离出来。借助多元稳定分布的分类与对称矩阵空间上的调和分析，我们证明了此类矩阵的稳定矩阵系综在矩阵表示、其特征值及其对角元方面的分类定理。此外，我们确定了其吸引域的充分必要条件。为说明我们的发现，我们讨论了例如椭圆不变随机矩阵系综和P\'olya系综，后者在矩阵卷积中扮演着特殊角色。作为副产品，我们将埃尔米特矩阵上的导数原理推广到一般的缓增分布。该原理建立了随机矩阵特征值联合概率密度与对角元之间的关系。