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}
1 Answer
If $m,n \to \infty$ with $\frac{m}n \to \lambda \in (0,\infty)$ (let's suppose that $n$ depends on $m \in\mathbb{N}$), and if you are saying that each $\mathbf{B} = \mathbf{B}_m$ is an $m\times n$ random matrix whose entries are i.i.d. drawn from a fixed (across $n$) probability distribution $\mu$ with zero mean and unit variance, then the behavior of $$ \frac1m \mathbf{B}\mathbf{B}^\dagger $$ is known to be potentially very far from $\mathbf{I}_m$, and even much worse its inverse. This is the content of the Marcenko-Pastur law, https://en.wikipedia.org/wiki/Marchenko%E2%80%93Pastur_distribution
On the other hand, products of matrices of the type you described appear in multivariate Fisher statistics, which is covered in Chapter 2.5 in a great book by Yao et. al., Large sample covariance matrices and high-dimensional data analysis. A revised edition can be found here: https://www.researchgate.net/publication/272093238_Large_Sample_Covariance_Matrices_and_High-Dimensional_Data_Analysis
