  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.
