Recently, we have classified Hermitian random matrix ensembles that are invariant under the conjugate action of the unitary group and stable with respect to matrix addition. Apart from a scaling and a shift, the whole information of such an ensemble is encoded in the stability exponent determining the ``heaviness'' of the tail and the spectral measure that describes the anisotropy of the probability distribution. In the present work, we address the question how these ensembles can be generated by the knowledge of the latter two quantities. We consider a sum of a specific construction of identically and independently distributed random matrices that are based on Haar distributed unitary matrices and a stable random vectors. For this construction, we derive the rate of convergence in the supremums norm and show that this rate is optimal in the class of all stable invariant random matrices for a fixed stability exponent. As a consequence we also give the rate of convergence in the total variation distance.

翻译：最近，我们对在酉群共轭作用下保持不变且关于矩阵加法稳定的Hermitian随机矩阵系综进行了分类。除缩放和平移外，此类系综的全部信息由两个量决定：决定尾部“重性”的稳定性指数，以及描述概率分布各向异性的谱测度。本文探讨如何通过已知的这两个量生成这些系综。我们考虑一类基于Haar分布酉矩阵与稳定随机向量的独立同分布随机矩阵特定构造之和。针对该构造，我们推导出上确界范数下的收敛速率，并证明该速率在给定稳定性指数的所有稳定不变随机矩阵类中是最优的。作为推论，我们还给出了全变差距离下的收敛速率。