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Theorem 1 and Le Cam’s first lemma (van der Vaart (1998), p.88) imply that the joint distributions of λ$_{1}$, ..., λ$_{m}$ (as well as those of µ$_{1}$, ..., µ$_{m}$$_{−}$$_{1}$ ) under the null and under the alternative are mutually contiguous for any h ∈ [0 $_{,}$√$_{c}$$_{)}$ $^{r}$. Along with Le Cam’s third lemma (van der Vaart (1998), p.90), this can be used to study the “local” powers of tests detecting signals in noise. 

Let β$_{λ}$ ( h ) and β$_{µ}$ ( h ) be the asymptotic powers of the asymptotically most powerful λ - and µ -based tests of size α of the null h = 0 against a point alternative h = ( h$_{1}$, ..., h$_{r}$ ) with h$_{j}$ < √$_{c, j}$ = 1 , ..., r . We have 

Theorem 2 Let Φ denote the standard normal distribution function. Then, 

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The theorem implies in particular that detection of signals corresponding to covariance spikes of sizes well below the phase transition threshold is possible with high probability. Consider for example the case where the number of observations equals the dimensionality of data so that c = 1 , the number of signals under the 

