Complex Gaussian Distribution Property Assume that we have a matrix $H$ of size $K \times M$ that has its entries following complex Gaussian distribution $\mathcal{CN}(0,1)$. One paper states the following properties
$$ \lim_{M/K \rightarrow \infty} HH^{*} \propto I_K$$
My question is that why we need the condition $M/K \rightarrow \infty$? I wonder that is it still correct if $M \approx K$?
 A: If $H$ is a Gaussian random matrix, then matrices of the form $HH^\dagger$ are called Wishart matrices. They form the Wishart ensemble or the Laguerre ensemble of random matrices. You may want to look that up.
The average value of $(HH^\dagger)_{ij}$ is $\langle(HH^\dagger)_{ij}\rangle=\sum_{m=1}^M\langle H_{im}H^*_{jm}\rangle=M\delta_{ij}$.
Its variance is $\langle (HH^\dagger)_{ij}^2\rangle-\langle (HH^\dagger)_{ij}\rangle^2=\sum_{m,n=1}^M\langle H_{im}H^*_{jm}H_{jn}H^*_{in}\rangle-(\sum_{m=1}^M\langle H_{im}H^*_{jm}\rangle)^2=M^2\delta_{ij}+M-M^2\delta_{ij}=M$
Therefore, the diagonal elements of $\frac{1}{M}HH^\dagger$ have average $1$ with standard deviation $1/\sqrt{M}$. The off-diagonal elements have average $0$ with standard deviation $1/\sqrt{M}$.
Now, if $K\to \infty$ as well, it is not correct to say that the matrix as a whole converges to the identity, because its behaviour depends on all its $K^2$ elements. For example, its eigenvalues will NOT become all equal (eigenvalues of random matrices repel one another).
