On the rate of convergence of the martingale central limit theorem in Wasserstein distances

TL;DR

This paper quantifies the convergence rate of the martingale central limit theorem in Wasserstein distances, providing bounds based on Lyapunov's coefficients and variance fluctuations, with optimality results.

Contribution

It introduces explicit bounds for the convergence rate in Wasserstein distances for martingales, extending previous results to a broader integrability range and establishing optimality of the bounds.

Findings

Wasserstein-1 bound is optimal up to a constant

Bounds depend on Lyapunov's coefficients and variance fluctuations

Applicable to martingales with wide integrability range

Abstract

For martingales with a wide range of integrability, we will quantify the rate of convergence of the central limit theorem via Wasserstein distances of order $r$, $1\le r\le 3$. Our bounds are in terms of Lyapunov's coefficients and the $\mathscr L^{r/2}$ fluctuation of the total conditional variances. We will show that our Wasserstein-1 bound is optimal up to a multiplicative constant.