ψ ( T ) is almost invariant . Moreover, since the conditions of Lemma 6.5.1 (same reference) hold, this test is invariant under the group ˜ G p (acting on T ). Since the ordered m -tuple λ$_{1}$, ..., λ$_{m}$ of the eigenvalues of 1 $_{n}$T = 1 $_{n}$XX ′ is maximal invariant for ˜ G $_{p}$, and since any invariant statistic is a measurable function of a maximal invariant one, ψ ( T ) must be λ -measurable. Hence, ψ ( T ) is a λ -measurable test and has the same power function as φ ( X ) , as was to be shown. The existence of a µ -measurable test with the same power function as that of a test φ ( X ) invariant under left orthogonal transformations and multiplication by non-zero constants is established similarly. □ References [1] Bai, Z.D. and J.W. Silverstein (2004) “CLT for Linear Spectral Statistics of Large-Dimensional Sample Covariance Matrices”, Annals of Probability 32, 553-605. [2] Bai, Z.D., Jiang, D., Yao, J.F., and S. Zheng (2009) “Corrections to LRT on Large-dimensional Covariance Matrix by RMT”, Annals of Statistics 37, 3822-3840. [3] Baik, J., Ben Arous, G. and S. Peche. (2005) “Phase transition of the largest eigenvalue for non-null complex sample covariance matrices” Annals of Prob- ability 33, 1643–1697. [4] Baik, J. and J.W. Silverstein. (2006) “Eigenvalues of large sample covariance matrices of spiked population models”. Journal of Multivariate Analysis 97, 1382–1408. [5] Berthet, Q. and Rigollet, P. (2012). Optimal detection of sparse principal components in high dimension. arXiv:1202.5070 40