# Asymptotic Analysis of higher-order scattering transform of Gaussian processes

Gi Ren Liu, Yuan Chung Sheu, Hau Tieng Wu

## 摘要

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.

原文 English 48 Electronic Journal of Probability 27 https://doi.org/10.1214/22-EJP766 Published - 2022

• 統計與概率
• 統計、概率和不確定性