NORMAL APPROXIMATION AND CONCENTRATION OF SPECTRAL PROJECTORS OF SAMPLE COVARIANCE

Let X, X.,..., Xn be i.i.d. Gaussian random variables in a separable Hubert space . with zero mean and covariance operator . = E(X . X), and let $\hat \Sigma : = {n^{ - 1}}\sum\nolimits_{j = 1}^n {\left( {{X_j} \otimes {X_j}} \right)} $ be the sample (empirical) covariance operator based on(X.,..., Xn). Denote by Pr the spectral projector of .. corresponding to its rth eigenvalue .r and by P.r the empirical counterpart of Pr. The main goal of the paper is to obtain tight bounds on $\matrix{\sup } \\ {x \in {\Cal R}} \\ \endmatrix \left| {{\Bbb P}\left\{ {\frac{{\left\| {{{\hat P}_r} - {P_r}} \right\|_2^2 - {\Bbb E}\left\| {{{\hat P}_r} - {P_r}} \right\|_2^2}}{{Va{r^{1/2}}\left( {\left\| {{{\hat p}_r} - {P_r}} \right\|_2^2} \right)}} \leqslant x} \right\} - \phi \left( x \right)} \right|$ where .·. denotes the Hilbert-Schmidt norm and . 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 . defined as $r\left( \Sigma \right) = \frac{{tr\left( \Sigma \right)}}{{{{\left\| \Sigma \right\|}_\infty }}}$ where tr(Z) is the trace of . and .... is its operator norm, as well as another parameter characterizing the size of Var $\left( {\left\| {{{\hat P}_r} - {P_r}} \right\|_2^2} \right)$. Other results include nonasymptotic bounds and asymptotic representations for the mean squared Hilbert-Schmidt norm error ${\Bbb E}\left\| {{{\hat P}_r} - {P_r}} \right\|_2^2$ and the variance Var $\left( {\left\| {{{\hat P}_r} - {P_r}} \right\|_2^2} \right)$ concentration inequalities for $\left\| {{{\hat P}_r} - {P_r}} \right\|_2^2$ around its expectation.