Asymptotics and concentration for sample covariance

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 Gaussian concentration. We then discuss several asymptotic and concentration results for spectral projectors of sample covariance in the case when the “effective rank” is large, but it is smaller than the sample size. These results include, in particular, asymptotic normality of bilinear forms of empirical spectral projectors and of squared Hilbert–Schmidt norms of their deviations from the spectral projectors of the true covariance.
Most of the results are joint with Karim Lounici