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.



Moment generating function of spectral norm of iid N(0,1) data matrix
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](https://projecteuclid.org/euclid.aos/1009210544). For finite $N$ results, Vershynin's [review](https://arxiv.org/abs/1011.3027v7) is a gold mine, see e.g. Corollary 5.35 therein. 


