Normal approximation and concentration of spectral projectors of sample covariance

Let $X,X_1,\dots ,X_n$ be i.i.d. Gaussian random variables in a separable Hilbert space $\mathbb{H}$ with zero mean and covariance operator $\mathrm{\Sigma }=\mathbb{E}(X\otimes X),$ and let $\hat \Sigma:=n^{-1}\sum_{j=1}^n (X_j\otimes X_j)$bethesample(empirical)covarianceoperatorbasedon$(X_1,\dots, X_n).$Denoteby$P_r$thespectralprojectorof$\Sigma$correspondingtoits$r$-theigenvalue$\mu_r$andby$\hat P_r$ the empirical counterpart of $P_r\mathrm{.}$ The main goal of the paper is to obtain tight bounds on $$\underset{x\in \mathbb{R}}{\sup }\mid \mathbb{P}\{\frac{\mathrm{\parallel }{\widehat{P}}_r-P_r{\mathrm{\parallel }}_2^2-\mathbb{E}\mathrm{\parallel }{\widehat{P}}_r-P_r{\mathrm{\parallel }}_2^2}{{\mathrm{V}\mathrm{a}\mathrm{r}}^{1\mathrm{/}2}(\mathrm{\parallel }{\widehat{P}}_r-P_r{\mathrm{\parallel }}_2^2)}\le x\}-\mathrm{\Phi }(x)\mid ,$$ where $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_2$ denotes the Hilbert–Schmidt norm and $\mathrm{\Phi }$ 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 $\mathrm{\Sigma }$ defined as ${\bf r}(\Sigma)=\frac{{\rm tr}(\Sigma)}{|\Sigma|{\infty}},$where${\rm tr}(\Sigma)$isthetraceof$\Sigma$and$|\Sigma|{\infty}$ is its operator norm, as well as another parameter characterizing the size of $\mathrm{V}\mathrm{a}\mathrm{r}(\mathrm{\parallel }{\widehat{P}}_r-P_r{\mathrm{\parallel }}_2^2)\mathrm{.}$ Other results include non-asymptotic bounds and asymptotic representations for the mean squared Hilbert–Schmidt norm error $\mathbb{E}\mathrm{\parallel }{\widehat{P}}_r-P_r{\mathrm{\parallel }}_2^2$ and the variance $\mathrm{V}\mathrm{a}\mathrm{r}(\mathrm{\parallel }{\widehat{P}}_r-P_r{\mathrm{\parallel }}_2^2),$ and concentration inequalities for $\mathrm{\parallel }{\widehat{P}}_r-P_r{\mathrm{\parallel }}_2^2$ around its expectation.