Hello everyone, I'm interested in doing statistical tests on properties of principal components, but none of the literature I've found so far seems quite right for my purposes. Many articles present asymptotic results like "the distribution of components converges to normal", but I haven't found any work describing convergence rates or other properties of the approximation in detail.

In particular, I've run a number of numerical experiments with differently distributed data which seem to indicate that components contributing only a small amount of variance gravitate towards some platykurtic distribution with parameters largely independent of the inital distribution - even in the finite case. Does anyone have a hint where I might look for helpful results?