A Sharp Rate of Convergence in the Functional Central Limit Theorem with Gaussian Input

TL;DR

This paper investigates the convergence rate of the functional central limit theorem for stationary Gaussian processes in the Wasserstein-1 metric, establishing precise bounds and revealing a faster convergence rate than in other metrics.

Contribution

It provides the first matching upper and lower bounds for the convergence rate of the functional CLT in the Wasserstein-1 metric for Gaussian processes.

Findings

Convergence rate is slightly faster in Wasserstein-1 than in Lévy-Prokhorov metric.

Established matching upper and lower bounds for the convergence rate.

Highlights the non-trivial nature of the functional CLT for Gaussian processes.

Abstract

When the underlying random variables are Gaussian, the classical Central Limit Theorem (CLT) is trivial, but the functional CLT is not. The objective of the paper is to investigate the functional CLT for stationary Gaussian processes in the Wasserstein-1 metric on the space of continuous functions. Matching upper and lower bounds are established, indicating that the convergence rate is slightly faster than in the L\'{e}vy-Prokhorov metric.