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Proposition 4 Let Φ denote the standard normal distribution function. Then, 

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Figure 1 shows the asymptotic power envelopes β$_{λ}$ ( h ) and β$_{µ}$ ( h ) as functions of h$_{1}$$_{/}$√$_{c}$ and h$_{2}$$_{/}$√$_{c}$ when h = ( h$_{1}$, h$_{2}$ ) is two-dimensional. 

It is important to realize that the asymptotic power envelopes derived in Propo- sition 4 are valid not only for λ - and µ -based tests but also for any test invariant under left orthogonal transformations of the observations ( X ↦→ QX, where Q is a p × p orthogonal matrix), and for any test invariant under multiplication by any non-zero constant and left orthogonal transformations of the observations ( X ↦→ aQX, where a ∈ R + 0 and Q is a p × p orthogonal matrix), respectively. 

