The asymptotic distribution for the principal component roots under local alternatives to multiple population roots is derived. The asymptotic theory assumes the estimate of the population covariance or scatter matrix to be asymptotically normal and to possess certain invariance properties. These assumptions are satisfied for the affine-invariant $M$-estimates of scatter for an elliptical distribution. The local alternative framework is used in deriving a local power function for the test for subsphericity.
"The Asymptotic Distribution of Principal Component Roots Under Local Alternatives to Multiple Roots." Ann. Statist. 11 (4) 1232 - 1242, December, 1983. https://doi.org/10.1214/aos/1176346336