A simple proof of almost sure convergence for the largest singular value of a product of Gaussian matrices
Thiziri Nait Saada, Alireza Naderi
TL;DR
This work analyzes the product of $m$ independent Gaussian matrices $\mathbf{W}=\mathbf{W}_1\cdots\mathbf{W}_m$ with entries $\mathcal{N}(0,n^{-1/2})$ and shows that the largest squared singular value converges almost surely to the endpoint $u_m=\frac{(m+1)^{m+1}}{m^{m}}$ of the limiting spectrum, which is supported by the Fuss–Catalan law. Building on Geman's moment-based approach, the authors provide a straightforward, self-contained proof that avoids free probability by bounding high moments of the largest eigenvalue via a finite-$n$ moment calculation. Central to the argument is a non-asymptotic expression for the $k$-th moment of the empirical spectral distribution, together with Stirling-number identities, which yields an explicit bound $\mathbb{E}((s_1^2)^k) \lesssim \frac{n}{k^{3/2}} u_m^{k}$. The probabilistic step uses the Borel–Cantelli lemma with a $k_n=\lceil w\log n\rceil$ growth to guarantee summability for appropriate $w$, establishing the almost-sure convergence and extending the classical $m=1$ soft edge result to product ensembles.
Abstract
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 $u_m = \frac{{(m+1)}^{m+1}}{m^m}$. 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. $s_1^2(\mathbf{W}) \xrightarrow{a.s.} u_1$. Herein, we present a proof for the general case $s_1^2(\mathbf{W}) \xrightarrow{a.s.} u_m$ for $m\geq 1$. Although we do not claim novelty for our result, the proof is simple and does not require familiarity with modern techniques of free probability.