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

Abstract

Let $m\ge 1$ 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 $u_m=\frac{{(m+1)}^{m+1}}{m^m}$. This generalises the well-known case of $m=1$, corresponding to the Marchenko-Pastur distribution for square matrices. Moreover, for $m=1$, it was first shown by Geman (1980) that the largest squared singular value almost surely converges to the right endpoint (the so-called ``soft edge'') of the support, i.e. $s_1^2(\mathbf{W})\stackrel{a.s.}{\to }{u}_1$. Herein, we present a proof for the general case $s_1^2(\mathbf{W})\stackrel{a.s.}{\to }{u}_m$ for $m\ge 1$. Although we do not claim novelty for our result, the proof is simple and does not require familiarity with modern techniques of free probability.