Normal approximation and concentration of spectral projectors of sample covariance

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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⁻¹∑ⱼ₌₁ⁿ (Xⱼ⊗ Xⱼ) be the sample (empirical) covariance operator based on (X₁,, Xₙ). Denote by Pᵣ the spectral projector of Σ corresponding to its r-th eigenvalue μᵣ and by Pᵣ the empirical counterpart of Pᵣ. The main goal of the paper is to obtain tight bounds on x∈ R |P{ Pᵣ-Pᵣ₂²-E Pᵣ-Pᵣ₂² Var¹/²(Pᵣ-Pᵣ₂²)≤ 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)Σ∞, 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-Pᵣ₂²). Other results include non-asymptotic bounds and asymptotic representations for the mean squared Hilbert--Schmidt norm error E Pᵣ-Pᵣ₂² and the variance Var(Pᵣ-Pᵣ₂²), and concentration inequalities for Pᵣ-Pᵣ₂² around its expectation.