  We analyze the joint probability distribution on the lengths of the vectors
of hidden variables in different layers of a fully connected deep network, when
the weights and biases are chosen randomly according to Gaussian distributions,
and the input is in $\{ -1, 1\}^N$. We show that, if the activation function
$\phi$ satisfies a minimal set of assumptions, satisfied by all activation
functions that we know that are used in practice, then, as the width of the
network gets large, the `length process' converges in probability to a length
map that is determined as a simple function of the variances of the random
weights and biases, and the activation function $\phi$. We also show that this
convergence may fail for $\phi$ that violate our assumptions.
