We will discuss recent moment bounds and concentration inequalities for sample covariance operators based on a sample of n i.i.d. Gaussian random variables taking values in an infinite dimensional space. These bounds show that the size of the operator norm of the deviation of sample covariance from the true covariance can be completely characterized by two parameters: the operator norm of the true covariance and its so called "effective rank". These results rely on Talagrand's generic chaining bounds and on…