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. 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.

中文翻译：

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移动平均模量小波变换及其变体的高斯近似


解析小波变换的复模量的移动平均值为小时间偏移和变形的信号提供了稳健的时间尺度表示。在这项工作中，我们通过 Malliavin 微积分和组合技术推导出了这种表示的稳态高斯过程的 Wiener 混沌扩展。扩展使我们能够获得两个长距离依赖的高斯过程的时间尺度表示之间的 Wasserstein 距离的下限（以 Hurst 指数表示）。此外，我们应用展开来建立平滑 Wasserstein 距离和 Kolmogorov 距离之间的上限，这些距离是从时间尺度表示中得出的随机向量与其正常对应物的分布之间的。值得一提的是，展开由无限 Wiener 混沌组成，随着 Wiener 混沌阶数的增加，投影系数缓慢收敛为零。我们为这些分布距离提供了一个有理衰减上限，其速率取决于复小波系数振幅的非线性变换。