Asymptotic Analysis of higher-order scattering transform of Gaussian processes

Gi Ren Liu, Yuan Chung Sheu, Hau Tieng Wu

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Article number48
JournalElectronic Journal of Probability
StatePublished - 2022


  • Malliavin calculus
  • Stein’s method
  • Wiener-Itô decomposition
  • scaling limits
  • scattering transform
  • wavelet transform

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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