Let $A$ be a gaussian matrix of size $d \times n$ where all the coefficients are drawn i.i.d. from $ \mathcal{N}(0, 1)$ and denote by $s_{\text{max}}$ its largest singular value.

Theorem 2.6 of http://www-personal.umich.edu/~rudelson/papers/rv-ICM2010.pdf mentions that : $$ \mathbb{E} (s_{\text{max}}) \leq \sqrt{n}+\sqrt{d} $$ Hence : $$ \mathbb{E} (\frac{s_{\text{max}}}{\sqrt{n}}-1 ) \leq \sqrt{\frac{d}{n}} $$

I'm looking for a non-asymptotic upper-bound on $ s_{\text{max}}^2 $ that would look like : $$ \mathbb{E} (~| \frac{s_{\text{max}}^2}{n}-1 |~) \leq C~ \frac{d}{n} $$ C being a constant. This is the same as bounding $ \frac{AA^T }{n} - I $, and the Frobenius norm already gives a bound in order of $ \frac{d^2}{n} $.