Let $\mathbf{A}_{m\times n}$ and $\mathbf{B}_{m\times n}$ be two random i.i.d matrices with zero mean and unit variance. Then, are the following large-scale analysis true (m,n go to infinity with fixed ratio)?
\begin{align}
\mathbf{A}^\mathrm{H}(\mathbf{B}\mathbf{B}^\mathrm{H})^{-1}\mathbf{A}=\frac{1}{m}\mathbf{A}^\mathrm{H}(\frac{1}{m}\mathbf{B}\mathbf{B}^\mathrm{H})^{-1}\mathbf{A},
\end{align}
as $\frac{1}{m}\mathbf{B}\mathbf{B}^\mathrm{H}\rightarrow\mathbf{I}_m$, where $\mathbf{I}_m$ is an identity matrix, we have
\begin{align}
\frac{1}{m}\mathbf{A}^\mathrm{H}(\frac{1}{m}\mathbf{B}\mathbf{B}^\mathrm{H})^{-1}\mathbf{A}&\rightarrow\frac{1}{m}\mathbf{A}^\mathrm{H}\mathbf{A}\\
&=\frac{n}{m}\frac{1}{n}\mathbf{A}^\mathrm{H}\mathbf{A}\\
&\rightarrow \frac{n}{m}\mathbf{I}_n.
\end{align}