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Motivated by analyzing complicated and non-stationary time series, we study ageneralization of the scattering transform (ST) that includes broad neuralactivation functions, which is called neural activation ST (NAST). On thewhole, NAST is a transform that comprises a sequence of ``neural processingunits'', each of which applies a high pass filter to the input from theprevious layer followed by a composition with a nonlinear function as theoutput to the next neuron. Here, the nonlinear function models how a neurongets excited by the input signal. In addition to showing properties likenon-expansion, horizontal translational invariability and insensitivity tolocal deformation, the statistical properties of the second order NAST of aGaussian process with various dependence and (non-)stationarity structure andits interaction with the chosen high pass filters and activation functions areexplored and central limit theorem (CLT) and non-CLT results are provided.Numerical simulations are also provided. The results explain how NAST processescomplicated and non-stationary time series, and pave a way towards statisticalinference based on NAST under the non-null case.

Central and Non-central Limit Theorems arising from the Scattering Transform and its Neural Activation Generalization