h, ˜ h ∈ [0 $_{,}$√$_{c}$ − δ ] r , Theorem 3 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}$$^{, ..., µ$_{m}$}$− $_{1}$) under the null and under the alternative are mutually contiguous for any h ∈ [0 $_{,}$√$_{c}$$_{)}$ $^{r}$. By applying Le Cam’s third lemma (van der Vaart (1998), p.90), we can study the “local” powers of tests detecting signals in noise. The requirement that h$_{j}$ be positive under alternatives corresponds to situations where the signals contained in the data are independent from the noise. If dependence between the signals and the noise is allowed, one might consider two-sided alternatives of the form H$_{1}$ : h$_{j}$ = 0 for some j . Values of h$_{j}$ between − 1 and 0 correspond to alternatives under which the noise variance is reduced along certain directions. In view of Proposition 2, it should not be difficult to generalize Theorem 3 to the case of fully ( h$_{j}$ = 0 , all j ’s) or partially ( h$_{j}$ = 0 , some j ’s) two-sided alternatives. This problem will not be discussed here, and is left for future research. Denote by β$_{λ}$ ( h ) and β$_{µ}$ ( h ) , respectively, the asymptotic powers of the asymp- totically most powerful λ - and µ -based tests of size α of the null h = 0 against a point alternative h = ( h$_{1}$, ..., h$_{r}$ ) = 0 with h$_{j}$ < √$_{c, j}$ = 1 , ..., r . As functions of h, β$_{λ}$ and β$_{µ}$ are called the asymptotic power envelopes . 12 3 Asymptotic power analysis Cov ( L $_{λ}$( h ) , L λ ( ˜ h )) = − 1 2 r ∑ i,j =1 ln ( 1 − h$_{i}$ ˜ h$_{j}$ c ) , and (12) Cov L $_{µ}$( h ) , L µ ˜ h = − 1 2 r i,j $_{=1}$( ln ( 1 − h$_{i}$ ˜ h$_{j}$ c ) + h$_{i}$ ˜ h$_{j}$ c ) . (13)