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Given finite i.i.d.~samples in a Hilbert space with zero mean and trace-classcovariance operator $\Sigma$, the problem of recovering the spectral projectorsof $\Sigma$ naturally arises in many applications. In this paper, we considerthe problem of finding distributional approximations of the spectral projectorsof the empirical covariance operator $\hat \Sigma$, and offer a dimension-freeframework where the complexity is characterized by the so-called relative rankof $\Sigma$. In this setting, novel quantitative limit theorems and bootstrapapproximations are presented subject only to mild conditions in terms ofmoments and spectral decay. In many cases, these even improve upon existingresults in a Gaussian setting.

Quantitative limit theorems and bootstrap approximations for empirical spectral projectors