Open Access
2022 Asymptotic analysis of higher-order scattering transform of Gaussian processes
Gi-Ren Liu, Yuan-Chung Sheu, Hau-Tieng Wu
Author Affiliations +
Electron. J. Probab. 27: 1-27 (2022). DOI: 10.1214/22-EJP766


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.

Funding Statement

This work benefited from support of the National Center for Theoretical Science (NCTS, Taiwan) and the Ministry of Science and Technology (MOST, Taiwan).


The authors would like to thank the editors for handling their paper and the anonymous reviewers for their valuable comments. Their comments have significantly improved the quality of their paper. G. R. Liu’s work was supported by the Ministry of Science and Technology under Grant 110-2628-M-006-003-MY3. Y. C. Sheu’s work was supported by the Ministry of Science and Technology under Grant 110-2115-M-A49 -007.


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Gi-Ren Liu. Yuan-Chung Sheu. Hau-Tieng Wu. "Asymptotic analysis of higher-order scattering transform of Gaussian processes." Electron. J. Probab. 27 1 - 27, 2022.


Received: 12 August 2021; Accepted: 17 March 2022; Published: 2022
First available in Project Euclid: 14 April 2022

MathSciNet: MR4408124
Digital Object Identifier: 10.1214/22-EJP766

Primary: 60G60 , 60H05 , 62M15
Secondary: 35K15

Keywords: Malliavin calculus , scaling limits , scattering transform , Stein’s method , wavelet transform , Wiener-Itô decomposition

Vol.27 • 2022
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