  We study products of random matrices in the regime where the number of terms
and the size of the matrices simultaneously tend to infinity. Our main theorem
is that the logarithm of the $\ell_2$ norm of such a product applied to any
fixed vector is asymptotically Gaussian. The fluctuations we find can be
thought of as a finite temperature correction to the limit in which first the
size and then the number of matrices tend to infinity. Depending on the scaling
limit considered, the mean and variance of the limiting Gaussian depend only on
either the first two or the first four moments of the measure from which matrix
entries are drawn. We also obtain explicit error bounds on the moments of the
norm and the Kolmogorov-Smirnov distance to a Gaussian. Finally, we apply our
result to obtain precise information about the stability of gradients in
randomly initialized deep neural networks with ReLU activations. This provides
a quantitative measure of the extent to which the exploding and vanishing
gradient problem occurs in a fully connected neural network with ReLU
activations and a given architecture.
