Let $X_n\in\mathbb{R}^{n\times m}$ be a matrix whose entries are i.i.d. random variables with zero mean and variance $\sigma^2$. Let $m$ be a fixed integer and $\|\cdot\|$ denote the 2-norm of a matrix. I would like to prove that $$ \left\|\frac{1}{n} X_n^\top X_n -\sigma^2 I\right\| \overset{P}{\to} 0 \ \ \ \text{ as }\ \ \ n\to\infty $$ where $\overset{P}{\to} 0$ denotes convergence in probability.
I believe that this should be a well-known fact, which should follow from the fact that high-dimensional random vectors are almost orthogonal as their dimension increases (see e.g. here). However, I couldn't find a reference with a rigorous proof. Hence, I would really appreciate any comment with pointers to the literature. Thanks!