  This paper deals with the asymptotic distribution of Wishart matrix and its
application to the estimation of the population matrix parameter when the
population eigenvalues are block-wise infinitely dispersed. We show that the
appropriately normalized eigenvectors and eigenvalues asymptotically generate
two Wishart matrices and one normally distributed random matrix, which are
mutually independent. For a family of orthogonally equivariant estimators, we
calculate the asymptotic risks with respect to the entropy or the quadratic
loss function and derive the asymptotically best estimator among the family. We
numerically show 1) the convergence in both the distributions and the risks are
quick enough for a practical use, 2) the asymptotically best estimator is
robust against the deviation of the population eigenvalues from the block-wise
infinite dispersion.
