Asymptotics and Concentration Bounds for Spectral Projectors of Sample Covariance

Let $X,X_1,\dots, X_n$ be i.i.d. Gaussian random variables with zero mean and covariance operator $\Sigma={\mathbb E}(X\otimes X)$ taking values in a separable Hilbert space ${\mathbb H}.$ Let ${\bf r}(\Sigma):=\frac{{\rm tr}(\Sigma)}{\|\Sigma\|_{\infty}}$ be the effective rank of $\Sigma$. Let $\hat \Sigma_n:=n^{-1}\sum_{j=1}^n (X_j\otimes X_j)$ be the sample (empirical) covariance operator based on $(X_1,\dots, X_n).$ The paper deals with a problem of estimation of spectral projectors of the covariance operator $\Sigma$ by their empirical counterparts, the spectral projectors of $\hat \Sigma_n$ (empirical spectral projectors). The focus is on the problems where both the sample size $n$ and the effective rank ${\bf r}(\Sigma)$ are large. This framework includes and generalizes well known high-dimensional spiked covariance models. Given a spectral projector $P_r$ corresponding to an eigenvalue $\mu_r$ of covariance operator $\Sigma$ and its empirical counterpart $\hat P_r,$ we derive sharp concentration bounds for the empirical spectral projector $\hat P_r$ in terms of sample size $n$ and effective dimension ${\bf r}(\Sigma).$ Building upon these concentration bounds, we prove the asymptotic normality of bilinear forms of random operators $\hat P_r -{\mathbb E}\hat P_r$ under the assumptions that $n\to \infty$ and ${\bf r}(\Sigma)=o(n).$ We also establish asymptotic normality of squared Hilbert--Schmidt norms $\|\hat P_r-P_r\|_2^2$ centered with their expectations and properly normalized. Other results include risk bounds for empirical spectral projectors $\hat P_r$ in Hilbert--Schmidt norm in terms of ${\bf r}(\Sigma),$ $n$ and other parameters of the problem, bounds on the bias ${\mathbb E}\hat P_r-P_r$ as well as a discussion of possible applications to statistical inference in high-dimensional principal component analysis.