The Product of Gaussian Matrices is Close to Gaussian

We study the distribution of the {\it matrix product} $G1 G2 \cdots Gr$ of $r$ independent Gaussian matrices of various sizes, where $Gi$ is $d{i-1} \times di$, and we denote $p = d0$, $q = dr$, and require $d1 = d{r-1}$. Here the entries in each $Gi$ are standard normal random variables with mean $0$ and variance $1$. Such products arise in the study of wireless communication, dynamical systems, and quantum transport, among other places. We show that, provided each $di$, $i = 1, \ldots, r$, satisfies $di \geq C p \cdot q$, where $C \geq C0$ for a constant $C0 > 0$ depending on $r$, then the matrix product $G1 G2 \cdots Gr$ has variation distance at most $\delta$ to a $p \times q$ matrix $G$ of i.i.d.\ standard normal random variables with mean $0$ and variance $\prod{i=1}^{r-1} di$. Here $\delta \rightarrow 0$ as $C \rightarrow \infty$. Moreover, we show a converse for constant $r$ that if $d_i < C' \max{p,q}^{{1/2}\min{p,q}{3/2}$} for some $i$, then this total variation distance is at least $\delta'$, for an absolute constant $\delta' > 0$ depending on $C'$ and $r$. This converse is best possible when $p=\Theta(q)$.