Let $X,X_1,\dots, X_n,\dots$ be i.i.d. centered Gaussian random variables in a separable Banach space $E$ with covariance operator $\Sigma:$ $$ \Sigma:E^{ \ast}\mapsto E,\ \ \Sigma u = { \mathbb E}\langle X,u\rangle, u\in E^{ \ast}. $$ The sample covariance operator $\hat \Sigma:E^{ \ast}\mapsto E$ is defined as $$ \hat \Sigma u := n^{ -1}\sum_{ j=1}^n \langle X_j,u\rangle X_j, u\in E^{ \ast}. $$ The goal of the paper is to obtain concentration inequalities and expectation bounds for the operator norm $\|\hat \Sigma-\Sigma\|$ of the deviation of the sample covariance operator from the true covariance operator. In particular, it is shown that $$ { \mathbb E}\|\hat \Sigma-\Sigma\|\asymp \|\Sigma\|\biggl(\sqrt{ \frac{ { \bf r}(\Sigma)}{ n}}\bigvee \frac{ { \bf r}(\Sigma)}{ n}\biggr), $$ where $$ { \bf r}(\Sigma):=\frac{ \Bigl({ \mathbb E}\|X\|\Bigr)^2}{ \|\Sigma\|}. $$ Moreover, under the assumption that ${ \bf r}(\Sigma)\lesssim n,$ it is proved that, for all $t\geq 1,$ with probability at least $1-e^{ -t}$ \begin{ align*} \Bigl|\|\hat\Sigma - \Sigma\|-{ \mathbb E}\|\hat\Sigma - \Sigma\|\Bigr| \lesssim \|\Sigma\|\biggl(\sqrt{ \frac{ t}{ n}}\bigvee \frac{ t}{ n}\biggr). \end{ align*}
Categories: math.PR