It is known that randomly initialized neural networks have Gaussian behaviour in the $n\to \infty$ limit. Unfortunately, this approximation can be poor for finite $n$ and gets worse as one goes deeper into the network. In this work, we develop a more accurate approximation that is \emph{slightly} non-Gaussian, which we call the para-Gaussian distribution and is constructed from the Hermite polynomials. We show how this para-Gaussian approximation can lead to a practical recursive method to approximate the deeper layers of the network. Finally, we apply this theory to linear neural networks (where things can be solved exactly) and to ReLU neural networks (where things can be approximated).

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