Consider the tests described in Examples 1, 2, 3, 4 and 5, and denote by β$_{J}$ ( h ) , β$_{LW}$ ( h ) , β$_{CLR}$ ( h ) , β$_{CM}$ ( h ) and β$_{TW}$ ( h ) their respective asymptotic powers at asymptotic level α. Proposition 6 The asymptotic power functions of the tests described in Exam- ples 1-5 are Formula (20) for β$_{CM}$ ( h ) directly follows from Proposition 2 of Cai and Ma (2012). The proof of the other formulae follows along the same lines as in the proof of Proposition 10 in OMH, and is omitted. Except for the Tracy-Widom tests of Example 4, all those asymptotic power functions are non-trivial. Figures 6 and 7 compare these power functions to the corresponding power envelopes for r = 2. Since John’s test is invariant with respect to orthogonal transformations and scalings of the data, Figure 6 compares β$_{J}$ ( h ) (solid line) to the power envelope β$_{µ}$ ( h ) (dotted line). The Ledoit-Wolf test, the “corrected” likelihood ratio test, and the Cai-Ma test are invariant with respect to orthogonal transformations of the data only, and Figure 7 thus compares the asymptotic power functions β$_{LW}$ ( h ) = β$_{CM}$ ( h ) and β$_{CLR}$ ( h ) (solid and dashed lines, respectively) to the power envelope β$_{λ}$ ( h ) (dotted line). Note that β$_{CLR}$ ( h ) depends on c . As c converges to one, β$_{CLR}$ ( h ) converges to α, which corresponds to the case of trivial power. As c converges to zero, β$_{CLR}$ ( h ) converges to β$_{LW}$ ( h ) = β$_{CM}$ ( h ). In Figure 7, we provide plots of β$_{CLR}$ ( h ) that correspond to c = 0 . 5 . 20 for any h = ( h$_{1}$, ..., h$_{r}$ ) = 0 such that h$_{j}$ ∈ [0 $_{,}$√$_{c}$$_{)}$ for j = 1 , ..., r. β$_{TW}$ ( h ) = α, (19) β$_{J}$ ( h ) = β$_{LW}$ ( h )= β$_{CM}$ ( h )=1 − Φ ( $_{Φ}$− $^{1}$(1 − α ) − 1 2 r ∑ j =1 h 2 j c ) , and (20) β$_{CLR}$ ( h ) = 1 − Φ ( $_{Φ}$− $^{1}$(1 − α ) − r ∑ j =1 h$_{j}$ − ln (1 + h$_{j}$ ) √ − 2 ln (1 − c ) − 2 c ) , (21)