  We study the distribution of the {\it matrix product} $G_1 G_2 \cdots G_r$ of
$r$ independent Gaussian matrices of various sizes, where $G_i$ is $d_{i-1}
\times d_i$, and we denote $p = d_0$, $q = d_r$, and require $d_1 = d_{r-1}$.
Here the entries in each $G_i$ 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 $d_i$, $i = 1, \ldots, r$, satisfies $d_i \geq C p
\cdot q$, where $C \geq C_0$ for a constant $C_0 > 0$ depending on $r$, then
the matrix product $G_1 G_2 \cdots G_r$ 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 $\prod_{i=1}^{r-1} d_i$. 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
$p=\Theta(q)$.
