Normal approximation and concentration of spectral projectors of sample covariance

Let X,X1, ⋯ Xn be i.i.d. Gaussian random variables in a separable Hilbert space H with zero mean and covariance operator Σ = E(X ⊗X), and let Σ := n -1Σnj =1(Xj ⊗Xj ) be the sample (empirical) covariance operator based on (X1, ⋯ Xn). Denote by Pr the spectral projector of Σ corresponding to its rth eigenvalue μr and by Pr the empirical counterpart of Pr. The main goal of the paper is to obtain tight bounds on sup xϵRP {|| Pr - Pr||22 -E|| Pr -Pr||22 Var1/2(|| Pr -Pr||22) = x} -φ(x), where || · ||2 denotes the Hilbert-Schmidt norm and φ is the standard normal distribution function. Such accuracy of normal approximation of the distribution of squared Hilbert-Schmidt error is characterized in terms of so-called effective rank of Σ defined as r(Σ) = tr(Σ) Σ8, where tr(Σ) is the trace of Σ and ||Σ||∞ is its operator norm, as well as another parameter characterizing the size of Var(|| Pr -Pr||22 ). Other results include nonasymptotic bounds and asymptotic representations for the mean squared Hilbert-Schmidt norm error E|| Pr - Pr||22 and the variance Var(|| Pr - Pr||22 ), and concentration inequalities for || Pr -Pr ||22 around its expectation.