Let $A$ be a $n\times m$ ($n<m$) random matrix with normal i.i.d. entries ($N(0,1)$).
Using the law of large numbers, it readily can be shown that as $n\to\infty$ $$ \frac{1}{n}A^TA\to I_m $$ where $I_m$ is $m\times m$ identity matrix. Indeed, the $(i,j)$ element of $A^TA$ is given by $$ \frac{1}{n}[A^TA]_{ij} = \frac{1}{n}\sum_{l=1}^nA_{li}A_{lj}\to\delta_{ij} $$ where $\delta_{ij}$ denoted the delta of Kronecker.
My question is regard the asymptotic behavior of $AA^T$, which is an $n\times n$ matrix with rank $m<n$. Namely, as $n\to\infty$, if we keep a fixed ratio $m/n\to\alpha\in(0,1)$, what can we say about the limit $$ \frac{1}{n}AA^T\to ? $$
This book chapter can answer your question. Particularly, section 5.3.