When scattering transform meets non-Gaussian random processes, a double scaling limit result

Abstract

Let T be a function of Hermite rank one and let ${\{G(t)\}}_{t\in \mathbb{R}}$ be a mean-square continuous stationary Gaussian process with long-range dependence. We explore the finite-dimensional distributions of the second-order scattering transform of the process $X={\{T(G(t))\}}_{t\in \mathbb{R}}$ when all the scale parameters go to infinity simultaneously. For frequently used wavelets, we find a constraint on the ratio of the scale parameters of the wavelet transform within the first and second layers such that the limit exists. The constraint is explicitly expressed in terms of the Hurst index of the long-range dependent inputs and the gap between the indices of the first and second non-zero coefficients in the Hermite expansion of the function T. Under the constraint on the ratio of the scale parameters, we prove that the rescaled second-order scattering transform converges in the finite-dimensional distribution sense to a chi process of degree one. The limiting process is expressed in terms of the Fourier transform of mother wavelet and the Hurst index of long-range dependence.