Normal approximation and concentration of spectral projectors of sample covariance (1504.07333v1)

Abstract: 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 $\Sigma={\mathbb E}(X\otimes X),$ and let $\hat \Sigma:=n^{-1}\sum_{j=1}^n (X_j\otimes X_j)$ be the sample (empirical) covariance operator based on $(X_1,\dots, X_n).$ Denote by $P_r$ the spectral projector of $\Sigma$ corresponding to its $r$-th eigenvalue $\mu_r$ and by $\hat P_r$ the empirical counterpart of $P_r.$ The main goal of the paper is to obtain tight bounds on $$ \sup_{x\in {\mathbb R}} \left|{\mathbb P}\left{\frac{|\hat P_r-P_r|2^2-{\mathbb E}|\hat P_r-P_r|_2^2}{{\rm Var}^{1/2}(|\hat P_r-P_r|_2^2)}\leq x\right}-\Phi(x)\right|, $$ where $|\cdot|_2$ denotes the Hilbert--Schmidt norm and $\Phi$ 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 $\Sigma$ defined as ${\bf r}(\Sigma)=\frac{{\rm tr}(\Sigma)}{|\Sigma|{\infty}},$ where ${\rm tr}(\Sigma)$ is the trace of $\Sigma$ and $|\Sigma|_{\infty}$ is its operator norm, as well as another parameter characterizing the size of ${\rm Var}(|\hat P_r-P_r|_2^2).$ Other results include non-asymptotic bounds and asymptotic representations for the mean squared Hilbert--Schmidt norm error ${\mathbb E}|\hat P_r-P_r|_2^2$ and the variance ${\rm Var}(|\hat P_r-P_r|_2^2),$ and concentration inequalities for $|\hat P_r-P_r|_2^2$ around its expectation.