0 2 4 6 0 0.5 1 h$_{1}$/c$^{1/2}$=0.3006 $_{2}$=(−log(1−h 2 $^{2}$/c))1/2 0 2 4 6 0 0.5 1 h$_{1}$/c$^{1/2}$=0.6683 $_{2}$=(−log(1−h 2 $^{2}$/c))1/2 0 2 4 6 0 0.5 1 h$_{1}$/c$^{1/2}$=0.9520 $_{2}$=(−log(1−h 2 $^{2}$/c))1/2 0 2 4 6 0 0.5 1 h$_{1}$/c$^{1/2}$=0.9995 $_{2}$=(−log(1−h 2 $^{2}$/c))1/2 Figure 6: Profiles of the asymptotic power of John’s test (solid lines) relative to the asymptotic power envelope β$_{µ}$ (dotted lines) for different values of h$_{1}$$_{/}$√$_{c}$ under the alternative; α = 0 . 05 . These comparisons show that, contrary to our LR tests (see Figures 2 and 3), all those tests either have trivial power α (the Tracy-Widom ones), or power functions that increase very slowly with h$_{1}$ and h$_{2}$, and lie very far below the corresponding power envelope. This paper extends Onatski, Moreira and Hallin’s (2011) (OMH) study of the power of high-dimensional sphericity tests to the case of multi-spiked alternatives. We derive the asymptotic distribution of the log-likelihood ratio process and use it to obtain simple analytical expressions for the maximal asymptotic power envelope and for the asymptotic powers of several tests proposed in the literature. These asymptotic powers turn out to be very substantially below the envelope. We propose the likelihood ratio test based on the data reduced to the eigenvalues of the sample covariance matrix. Our computations show that the asymptotic power 21 4 Conclusion