Asymptotic behavior of the distributions of eigenvalues for beta-Wishart ensemble under the dispersed population eigenvalues

We propose a Laplace approximation of the hypergeometric function with two matrix arguments expanded by Jack polynomials. This type of hypergeometric function appears in the joint density of eigenvalues of the beta-Wishart matrix for parameters (Formula presented.) where the matrix indicates the cases for reals, complexes, and quaternions, respectively. Using the Laplace approximations, we show that the joint density of the eigenvalues can be expressed using gamma density functions when population eigenvalues are infinitely dispersed. In general, for the parameter (Formula presented.) we also show that the distribution of the eigenvalue can be approximated by gamma distributions through broken arrow matrices. We compare approximated gamma distributions with empirical distributions by Monte Carlo simulation.