Concentration inequalities and moment bounds for sample covariance operators

Let X,X1, . . . , Xn, . . . be i.i.d. centered Gaussian random variables in a separable Banach space E with covariance operator: The sample covariance operator ∑ : E E is defined as The goal of the paper is to obtain concentration inequalities and expectation bounds for the operator norm of the deviation of the sample covariance operator from the true covariance operator. In particular, it is shown that E∑ -∑→ ∑ ∑r(∑)n∑ r(∑)n, wherer(∑) := (EX)2∑. Moreover, it is proved that, under the assumption that r(E) ≤ n, for all t ≤ 1, with probability at least 1 - e -t - ∑- ∑-M- ∑ ∑ ∑tnVtn, where M is either the median, or the expectation of On the other hand, under the assumption that r(∑) ≤ n, for all t ≤ 1, with probability at least 1- e -t -.