# Wald Statistics in high-dimensional PCA

In this note we consider PCA for Gaussian observations X_1,..., X_n with covariance Σ=∑_i λ_i P_i in the 'effective rank' setting with model complexity governed by r(Σ):=tr(Σ)/Σ. We prove a Berry-Essen type bound for a Wald Statistic of the spectral projector P̂_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 r(Σ) ≫√(n).

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