Let $X$ be a random $(p \times n)$-matrix with iid centered columns and suppose the entries of $X$ all have light tails (in a strong enough sense, for example sub-Gaussian). Are there any results bounding the tail probabilities of the largest singular value of $X$, i.e. the largest eigenvalue of the sample covariance matrix $XX^T$?
Usually, one assumes $X = \Sigma^{\frac{1}{2}} Z$ for some covariance matrix $\Sigma$ and a random $(p \times n)$-matrix $Z$ with iid entries. I am specifically not in such a setting and am interested if one can still give spectral bounds by only using information on the covariances of $X$ and the tail behaviour of the entries of $X$.
(PS. The asymptotic setting I am working with allows $n>>p$, if $n \sim p$ is too difficult.)
Any help is much appreciated!
