Then, L$_{p}$ ( h ; λ$_{p}$ ) = r ∏ j =1 (1 + h$_{j}$ $_{)}$− n p 2 ∫ ( p ) e p tr( D$_{p}$Q ′ Λ$_{p}$ Q $^{)}$(d Q ) and (3) O ( p ) L$_{p}$ ( h ; µ$_{p}$ ) = r ∏ j =1 (1 + h$_{j}$ $_{)}$− n p 2 ( n$_{p}$ 2 ) n p p 2 Γ ( n$_{p}$p 2 ) ∫ ∞ 0 x n p p 2 − 1 $_{e}$− n p 2 x ∫ O ( p ) e p x S p tr( D$_{p}$Q ′ Λ$_{p}$ Q $^{)}$(d Q )d x. (4) In the special case where r = 1 , the rank of the matrix D$_{p}$ equals one, and the integrals over the orthogonal group in (3) and (4) can be rewritten as integrals over a p -dimensional sphere. OMH show how such spherical integrals can be represented in the form of contour integrals, and apply Laplace approximation to these contour integrals to establish the asymptotic properties of L$_{p}$ ( h ; λ$_{p}$ ) and L$_{p}$ ( h ; µ$_{p}$ ) . In the r > 1 case, the integrals in (3) and (4) can be rewritten as integrals over a Stiefel manifold, the set of all orthonormal r -frames in R $^{p}$. Onatski (2012) obtains a generalization of the contour integral representation from spherical integrals to integrals over Stiefel manifolds. Unfortunately, the Laplace approximation method does not straightforwardly extend to that generalization, and we therefore propose an alternative method of analysis. The second-order asymptotic behavior, as p goes to infinity, of integrals of the form ∫ O ( p $_{)}$e p tr( DQ ′ Λ Q $^{)}$(d Q ) was analyzed in Guionnet and Maida (2005) (Theo- rem 3) for the particular case where D is a fixed matrix of rank one, Λ a deter- ministic matrix, and under the condition that the empirical distribution of Λ’s eigenvalues converges to a distribution function with bounded support. Below, we extend Guionnet and Maida’s approach to cases where D = D$_{p}$ has rank larger than one, and to the stochastic setting of this paper. We then use such an extension to derive the asymptotic properties of L$_{p}$ ( h ; λ$_{p}$ ) and L$_{p}$ ( h ; µ$_{p}$ ) . ˆ λ MP Let F p be the empirical distribution of λ$_{p}$$_{1}$, ..., λ$_{pp}$ , and denote by F p the 8