Gaussian Approximation for the Moving Averaged Modulus Wavelet Transform and its Variants

TL;DR

This paper derives a Wiener chaos expansion for the moving average of the complex modulus of the wavelet transform of Gaussian processes, providing bounds on distribution distances and insights into long-range dependence.

Contribution

It introduces a novel Wiener chaos expansion for this wavelet-based representation and establishes bounds on Wasserstein and Kolmogorov distances for Gaussian processes.

Findings

Provides a lower bound for Wasserstein distance based on Hurst indices.

Establishes an upper bound for distribution distances involving the chaos expansion.

Shows the projection coefficients decay slowly, affecting convergence rates.

Abstract

The moving average of the complex modulus of the analytic wavelet transform provides a robust time-scale representation for signals to small time shifts and deformation. In this work, we derive the Wiener chaos expansion of this representation for stationary Gaussian processes by the Malliavin calculus and combinatorial techniques. The expansion allows us to obtain a lower bound for the Wasserstein distance between the time-scale representations of two long-range dependent Gaussian processes in terms of Hurst indices. Moreover, we apply the expansion to establish an upper bound for the smooth Wasserstein distance and the Kolmogorov distance between the distributions of a random vector derived from the time-scale representation and its normal counterpart. It is worth mentioning that the expansion consists of infinite Wiener chaos and the projection coefficients converge to zero slowly as the order of the Wiener chaos increases, and we provide a rational-decay upper bound for these distribution distances, the rate of which depends on the nonlinear transformation of the amplitude of the complex wavelet coefficients.