A simple proof of almost sure convergence for the largest singular value of a product of Gaussian matrices

Understand

Let $m \geq 1$ and consider the product of $m$ independent $n \times n$ matrices $\mathbf{W} = \mathbf{W}_1 \dots \mathbf{W}_m$, each $\mathbf{W}_{i}$ with i.i.d.

• normalised $\mathcal{N}(0, n^{-1/2})$ entries.
• It is shown in Penson et al.
• (2011) that the empirical distribution of the squared singular values of $\mathbf{W}$ converges to a deterministic distribution compactly supported on $[0, u_m]$, where $u_m = \frac{{(m+1)}^{m+1}}{m^m}$.