  We analyze the scattering transform with the quadratic nonlinearity (STQN) of
Gaussian processes without depth limitation. STQN is a nonlinear transform that
involves a sequential interlacing convolution and nonlinear operators, which is
motivated to model the deep convolutional neural network. We prove that with a
proper normalization, the output of STQN converges to a chi-square process with
one degree of freedom in the finite dimensional distribution sense, and we
provide a total variation distance control of this convergence at each time
that converges to zero at an exponential rate. To show these, we derive a
recursive formula to represent the intricate nonlinearity of STQN by a linear
combination of Wiener chaos, and then apply the Malliavin calculus and Stein's
method to achieve the goal.
