Optimal Convergence Rates and One-Term Edgeworth Expansions for Multidimensional Functionals of Gaussian Fields

TL;DR

This paper develops advanced techniques using Malliavin calculus and Stein's method to precisely determine convergence rates and Edgeworth expansions for multidimensional Gaussian functionals, extending previous results to more complex, fluctuating covariance scenarios.

Contribution

It introduces a framework for exact asymptotic convergence analysis and Edgeworth expansions for smooth functions of Gaussian fields in multiple dimensions, allowing for covariance fluctuations.

Findings

Derived exact convergence speeds in multidimensional normal approximations.

Established one-term Edgeworth expansions for Gaussian functionals.

Applied methods to Brownian sheets, Toeplitz functionals, and the Breuer-Major theorem.

Abstract

We develop techniques for determining the exact asymptotic speed of convergence in the multidimensional normal approximation of smooth functions of Gaussian fields. As a by-product, our findings yield exact limits and often give rise to one-term generalized Edgeworth expansions increasing the speed of convergence. Our main mathematical tools are Malliavin calculus, Stein's method and the Fourth Moment Theorem. This work can be seen as an extension of the results of arXiv:0803.0458 to the multi-dimensional case, with the notable difference that in our framework covariances are allowed to fluctuate. We apply our findings to exploding functionals of Brownian sheets, vectors of Toeplitz quadratic functionals and the Breuer-Major Theorem.