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 $\Sigma,$ ${ \rm tr}(\Sigma)$ being the trace of $\Sigma$ and $\|\Sigma\|_{ \infty}$ being its operator norm. 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 bilinear forms of 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 ${ \bf r}(\Sigma)=o(n).$ In a special case of eigenvalues of multiplicity one, these results are rephrased as concentration bounds and asymptotic normality for linear forms of empirical eigenvectors. Other results include bounds on the bias ${ \mathbb E}\hat P_r-P_r$ and a method of bias reduction as well as a discussion of possible applications to statistical inference in high-dimensional principal component analysis.
Categories: math.ST stat.TH