process has mean E [ L $_{λ}$( h )] = 1 4 r ∑ i,j =1 ln (1 − h$_{i}$h$_{j}$/c ) and autocovariance function Cov ( L $_{λ}$( h ) , L $_{λ}$( ˜ h ) ) = − 1 2 ∑ r i,j $_{=1}$ln ( 1 − h$_{i}$ ˜ h$_{j}$/c ) . That convergence entails the weak convergence, in the Le Cam sense, of the h -indexed statistical experiments E m λ under which the eigenvalues λ$_{1}$, ..., λ$_{m}$ are observed, i.e. the statistical experiments with log-likelihood process { ln L$_{λ}$ ( h ) ; h ∈ [ 0 , ¯ h ] r } (see van der Vaart (1998), page 126). Although this limiting process is Gaussian, it is not a log-likelihood process of the Gaussian shift type, so that the statistical experiments E m λ under study are not locally asymptotically normal (LAN) ones. The weak convergence of E m λ implies, however, via Le Cam’s first lemma (see van der Vaart 1998, p.88), that the joint distributions of the normalized sample covariance eigenvalues under the null and under alternatives associated with h ∈ [0 $_{,}$√$_{c}$$_{) are mutually contiguous.}$ An asymptotic power envelope for λ -based tests of H$_{0}$ against H$_{1}$ can be con- structed 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 ( h ) ] , where Φ is the standard normal distribution function and W ( h ) = − 1 2 ∑ r i,j $_{=1}$ln (1 − h$_{i}$h$_{j}$/c ). As we explain in the paper, this asymptotic power envelope is valid not only for the λ -based tests, but also for all tests that are invariant under left orthogonal trans- formations of the data X$_{t}$, t = 1 , ..., n . Next, we consider previously proposed tests of sphericity and of the equality of the population covariance matrix to a given matrix . We focus on the tests studied in Ledoit and Wolf (2002), Bai et al (2009), and Cai and Ma (2012). We find that, in general, the asymptotic powers of those tests are substantially lower than the corresponding asymptotic power envelope value. In contrast, our computations for the case r = 2 show that the asymptotic powers of the λ - and µ -based likelihood ratio tests are close to the power envelope. 5