Wald Statistics in high-dimensional PCA

In this note we consider PCA for Gaussian observations $X_1,\dots ,X_n$ with covariance $\mathrm{\Sigma }={\sum }_i{\lambda }_i{P}_i$ in the ‘effective rank’ setting with model complexity governed by $$\mathbf{r}$(\Sigma):=$\text{tr}$(\Sigma)/| \Sigma |$. We prove a Berry-Essen type bound for a Wald Statistic of the spectral projector ${\widehat{P}}_r$. This can be used to construct non-asymptotic confidence ellipsoids and tests for spectral projectors $P_r$. Using higher order pertubation theory we are able to show that our Theorem remains valid even when $\mathbf{r}(\mathrm{\Sigma })\gg \sqrt{n}$.