  In this note we consider PCA for Gaussian observations $X_1,\dots, X_n$ with
covariance $\Sigma=\sum_i \lambda_i P_i$ 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 $\hat P_r$. 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
$\mathbf{r}(\Sigma) \gg \sqrt{n}$.
