Asymptotics of the right singular vectors as the number of rows diverge [duplicate]

Question

Write $X_m \in \mathbb{R}^{m \times n}$ as a Gaussian ensemble, so that $(X_m)_{ij} \sim \mathcal{N}(0, 1)$ are independent and identically distributed. Assume that $m \geq n$. Write $X_m = U_m \Sigma_m V_m^T$ as the singular value decomposition, where $V_m \in \mathbb{R}^{n \times n}$ are the right singular vectors.

What can be said about $V_m$ as $m \to \infty$? Is anyone aware of literature or techniques discussing this?

Answer 1

Thanks to the rotation-invariance of the Gaussian distribution, both $U_m$ and $V_m$ can be taken to be Haar-distributed orthogonal matrices in $\mathbb{R}^{m\times m}$ and $\mathbb{R}^{n\times n}$ respectively. This is true for all $m$ and $n$ and not just in the limit.