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

Let nS be an m× m matrix having the Wishart distribution Wm(n,Σ). For large n and simple latent roots of Σ, 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 Σ are simple.