Asymptotic Expansions for the Joint and Marginal Distributions of the Latent Roots of the Covariance Matrix

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.