contained in λ when p, n → ∞ so that p/n → c with 0 < c < ∞ . Our null hypothesis is H$_{0}$ : h$_{1}$ = ... = h$_{r}$ = 0 (no signals), and our alternative is H$_{1}$ : h$_{i}$ > 0 for some i = 1 , ..., r . The matrix V is left as an unspecified nuisance parameter. In this framework, signal detection tests can also be interpreted as tests of sphericity. We consider both cases of specified and unspecified σ $^{2}$. For the purpose of brevity, in the introduction, we will discuss only the case of specified σ 2 = 1. First, we study the likelihood ratio L ( h ; λ ) , defined as the ratio of the densities of λ cor- responding to unrestricted h and restricted h = 0, the densities being evaluated at the observed value of λ . We show that L ( h ; λ ) can be represented in the form of the determinant of an r × r matrix with entries equal to contour integrals of elementary functions. We use Laplace approximations to these contour integrals to show that for any ¯ h such that 0 < ¯ h < √$_{c,}$ with √$_{c}$ being the value of the phase transition threshold, the sequence of log-likelihood processes { ln L ( h ; λ ) ; h ∈ [0 , ¯ h ] $^{r}$} con- verges weakly to a Gaussian process 1 { L $_{λ}$( h ); h ∈ [ 0 , ¯ h ] $^{r}$} under the null hypothesis as n, p → ∞ . The limiting process has mean E [ L $_{λ}$( h )] = 1 2 ∑ r i,j $_{=1}$ln (1 − h$_{i}$h$_{j}$/c ) and autocovariance function Cov ( L $_{λ}$( h ) , L λ ( ˜ h )) = − r i,j $_{=1}$ln ( 1 − h$_{i}$ ˜ h$_{j}$/c ) . The established weak convergence of statistical experiments implies, via Le Cam’s first lemma (see van der Vaart 1998, p.88), that the joint distributions of the sam- ple covariance eigenvalues under the null and under alternatives associated with h ∈ [0 $_{,}$√$_{c}$$_{)}$ r are mutually contiguous. An asymptotic power envelope for eigenvalue-based tests of H$_{0}$ against H$_{1}$ can be constructed using the Neyman-Pearson lemma and Le Cam’s third lemma. We show that, for tests of size α, the maximum achievable asymptotic power against a point alternative h = ( h$_{1}$, ..., h$_{r}$ ) equals 1 − Φ [ Φ − $^{1}$(1 − α ) − $^{√$_{W}$}$] , where Φ is the standard normal distribution function and W = − ∑ r i,j $_{=1}$ln (1 − h$_{i}$h$_{j}$/c ). A $^{1}$Here the index λ in the notation $_{λ}$( h ) is used to distinguish the limiting log-likelihood $^{1}$Here the index λ in the notation L $_{λ}$( h ) is used to distinguish the limiting log-likelihood process in the case of specified σ 2 = 1 , from that in the case of unspecified σ $^{2}$, which we denote by L $_{µ}$( h ). 4