5.3 Proof of Proposition 4 C = √ W ( h $_{)Φ}$− $^{1}$(1 − α ) + m ( h ) , (56) To save space, we only derive the asymptotic power envelope for the relatively more difficult case of real-valued data and µ -based tests. According to the Neyman- Pearson lemma, the most powerful test of h = 0 against the simple alternative h = ( h$_{1}$, ..., h$_{r}$ ) is the test which rejects the null when L$_{p}$ ( h ; µ$_{p}$ ) is larger than a critical value C. It follows from Theorem 1 that, for such a test to have asymptotic size α , C must be where m ( h ) = 1 4 r ∑ i,j =1 ( ln ( 1 − h$_{i}$h$_{j}$ c ) + h$_{i}$h$_{j}$ c ) and W ( h ) = − 1 2 r i,j =1 ( ln ( 1 − h$_{i}$h$_{j}$ c ) + h$_{i}$h$_{j}$ c ) . 5.4 Invariance issues and Proof of Proposition 5 Now, according to Le Cam’s third lemma and Theorem 1, under h = ( h$_{1}$, ..., h$_{r}$ ) , ln L$_{p}$ ( h ; µ$_{p}$ ) d → N ( m ( h ) + W ( h ) , W ( h )) . The asymptotic power (15) follows. □ Before turning to the proof of Proposition 5, let us clarify the invariance issues in the problem under study. For basic definitions (invariant, maximal invariant, etc.), we refer to Chapter 6 of Lehmann and Romano (2005). Suppose that X is a p × n random matrix with vec ( X ) ∼ N (0 , I$_{n}$ ⊗ Σ). This model is clearly invariant under the group G $_{p}$, acting on R p × $^{n}$, of left-hand multi- plications by a p × p orthogonal matrix x ↦→ Qx, x ∈ R p × $^{n}$, Q ∈ O ( p ) ; so are the null hypothesis H$_{0}$ and the alternative H$_{1}$. Letting m = min ( n, p ) , the m -tuple λ ( X ) = ( λ$_{1}$, ..., λ$_{m}$ ) of non-zero eigenvalues of 1 $_{n}$XX ′ is clearly invariant under that group, since 1 $_{n}$xx ′ and 1 n ( Qx ) ( Qx $_{)}$′ = 1 $_{n}$Qxx ′ Q ′ share the same eigenvalues λ ( x ) for any orthogonal matrix Q and any matrix x ∈ R p × $^{n}$. However, λ ( X ) is 38