Let nS be an $m\times m$ matrix having the Wishart distribution $W_m(n,\Sigma)$. For large n and simple latent roots of $\Sigma$, it is known that the latent roots of S are asymptotically independently normal. In this paper an expansion, up to and including the terms of order $n^-1$, is given for the joint density function of the roots of S in terms of normal density functions. Expansions for the marginal distributions of the roots are also given, valid when the corresponding roots of $\Sigma$ are simple.
@article{1176343205,
author = {Muirhead, R. J. and Chikuse, Y.},
title = {Asymptotic Expansions for the Joint and Marginal Distributions of the Latent Roots of the Covariance Matrix},
journal = {Ann. Statist.},
volume = {3},
number = {1},
year = {1975},
pages = { 1011-1017},
language = {en},
url = {http://dml.mathdoc.fr/item/1176343205}
}
Muirhead, R. J.; Chikuse, Y. Asymptotic Expansions for the Joint and Marginal Distributions of the Latent Roots of the Covariance Matrix. Ann. Statist., Tome 3 (1975) no. 1, pp. 1011-1017. http://gdmltest.u-ga.fr/item/1176343205/