  This paper deals with the local asymptotic structure, in the sense of Le
Cam's asymptotic theory of statistical experiments, of the signal detection
problem in high dimension. More precisely, we consider the problem of testing
the null hypothesis of sphericity of a high-dimensional covariance matrix
against an alternative of (unspecified) multiple symmetry-breaking directions
(\textit{multispiked} alternatives). Simple analytical expressions for the
asymptotic power envelope and the asymptotic powers of previously proposed
tests are derived. These asymptotic powers are shown to lie very substantially
below the envelope, at least for relatively small values of the number of
symmetry-breaking directions under the alternative. In contrast, the asymptotic
power of the likelihood ratio test based on the eigenvalues of the sample
covariance matrix is shown to be close to that envelope. These results extend
to the case of multispiked alternatives the findings of an earlier study
(Onatski, Moreira and Hallin, 2011) of the single-spiked case. The methods we
are using here, however, are entirely new, as the Laplace approximations
considered in the single-spiked context do not extend to the multispiked case.
