Wasserstein-p Bounds via Cumulant-Based Edgeworth Expansion for $\alpha$-Mixing Random Fields

TL;DR

This paper extends Wasserstein-$p$ bounds to $ ext{alpha}$-mixing random fields, providing convergence rates to normality using cumulant-based Edgeworth expansions and a new graph approach.

Contribution

It introduces the first Wasserstein-$p$ bounds for dependent $ ext{alpha}$-mixing random fields with explicit convergence rates, using novel combinatorial and analytical techniques.

Findings

Convergence rate of $O(|T|^{-eta})$ in Wasserstein-$p$ distance.

Achieves rate $1/2$ under fast polynomial decay of mixing coefficients.

Develops a cumulant-based Edgeworth expansion for dependent fields.

Abstract

Recent progress has been made in establishing normal approximation bounds in terms of the Wasserstein-$p$ distance for i.i.d. and locally dependent random variables. However, for $p > 1$, no such results have been demonstrated for dependent variables under $\alpha$-mixing conditions. In this paper, we extend the Wasserstein-$p$ bounds to $\alpha$-mixing random fields. We show that, under appropriate conditions, the rescaled average of random fields converges to the standard normal distribution in the Wasserstein-$p$ distance at a rate of $O(|T|^{-\beta})$, where $|T|$ is the size of the index set, and $\beta \in (0, 1/2]$ depends on $p$, the dimension $d$ of the random fields, and the decay rate of the $\alpha$-mixing coefficients. Notably, $\beta = 1/2$ is achievable if the mixing coefficients decay at a sufficiently fast polynomial rate. Our results are derived through a carefully constructed cumulant-based Edgeworth expansion and an adaptation of recent developments in Stein's method. Additionally, we introduce a novel constructive graph approach that leverages combinatorial techniques to establish the desired expansion for general dependent variables.