Convergence rate analysis in limit theorems for nonlinear functionals of the second Wiener chaos

Abstract

This paper analyzes the distribution distance between random vectors from the analytic wavelet transform of squared envelopes of Gaussian processes and their large-scale limits. For Gaussian processes with a long-memory parameter below 1/2, the limit combines the second and fourth Wiener chaos. Using a non-Stein approach, we determine the convergence rate in the Kolmogorov metric. When the long-memory parameter exceeds 1/2, the limit is a chi-distributed random process, and the convergence rate in the Wasserstein metric is determined using multidimensional Stein’s method. Notable differences in convergence rate upper bounds are observed for long-memory parameters within (1/2,3/4) and (3/4,1).