Moment generating function of spectral norm of iid N(0,1) data matrix

Question

Let $W^{p\times p}$ be a normal data matrix with $W_{ij}$ i.i.d. $N(0,1)$. Are there any results on the evaluation, or upper bound for the Moment Generating Function of the spectral norm of W, that is, for $\lambda \in \mathbb{R}$, $E[\exp (\lambda ||W||)]$, where $||W|| = \sqrt{\lambda_1 (W^T W)}$ with $\lambda_1$ being the largest eigenvalue. If necessary, the result can be asymptotic in $p$.

Any hints, results, or references would be much appreciated. Thanks.

Answer 1

The asymptotic behavior of the operator norm of Wishart matrices ($W^TW$ in your notation) has been studied by Johnstone, see Theorem 1.1 in his paper. For finite $N$ results, Vershynin's review is a gold mine, see e.g. Corollary 5.35 therein.