2015/04/28 by Koltchinskii, Vladimir, Lounici, Karim · 4 citations
#62H12 #FOS: Mathematics #Statistics Theory (math.ST)
paper · doi:10.48550/arxiv.1504.07333
Let X,X1,…, Xn be i.i.d. Gaussian random variables in a separable Hilbert space \mathbb H with zero mean and covariance operator Σ=\mathbb 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 sup_x∈ \mathbb R |\mathbb P\\frac‖ Pr-Pr‖22-\mathbb E‖ Pr-Pr‖22\rm 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 \bf r(Σ)=\frac\rm tr(Σ)‖Σ‖∞, where \rm tr(Σ) is the trace of Σ and ‖Σ‖∞ is its operator norm, as well as another parameter characterizing the size of \rm Var(‖ Pr-Pr‖22). Other results include non-asymptotic bounds and asymptotic representations for the mean squared Hilbert--Schmidt norm error \mathbb E‖ Pr-Pr‖22 and the variance \rm Var(‖ Pr-Pr‖22), and concentration inequalities for ‖ Pr-Pr‖22 around its expectation.