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Let $X$ be a mean zero Gaussian random vector in a separable Hilbert space ${\mathbb H}$ with covariance operator $\Sigma:={\mathbb E}(X\otimes X).$ Let $\Sigma=\sum_{r\geq 1}\mu_r P_r$ be the spectral decomposition of $\Sigma$ with distinct eigenvalues $\mu_1>\mu_2> \dots$ and the corresponding spectral projectors $P_1, P_2, \dots.$ Given a sample $X_1,\dots, X_n$ of size $n$ of i.i.d.
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