CENTRAL AND NONCENTRAL LIMIT THEOREMS ARISING FROM THE SCATTERING TRANSFORM AND ITS NEURAL ACTIVATION GENERALIZATION

Motivated by the analysis of complicated time series, we examine a generalization of the scattering transform that includes broad neural activation functions. This generalization is the neural activation scattering transform (NAST). NAST comprises a sequence of "neural processing units," each of which applies a high pass filter to the input from the previous layer followed by a composition with a nonlinear function as the output to the next neuron. Here, the nonlinear function models how a neuron gets excited by the input signal. In addition to showing properties like nonexpansion, horizontal translational invariability, and insensitivity to local deformation, we explore the statistical properties of the second-order NAST of a Gaussian process with various dependence structures and its interaction with the chosen wavelets and activation functions. We also provide central limit theorem (CLT) and non-CLT results. Numerical simulations demonstrate the developed theorems. Our results explain how NAST processes complicated time series, paving a way toward statistical inference based on NAST for real-world applications.