Wasserstein-p Bounds via Cumulant-Based Edgeworth Expansion for $α$-Mixing Random Fields
Tianle Liu, Morgane Austern
TL;DR
This work extends Wasserstein-$p$ normal approximation bounds to $\alpha$-mixing random fields by developing a cumulant-based Edgeworth expansion and a novel constructive graph (genogram) framework to manage high-order dependence terms. Under moment and mixing conditions, including polynomial decay of the mixing coefficients, it establishes an Edgeworth expansion for $\mathbb{E}[h(W_n)]$ and shows $\mathcal{W}_p(\mathcal{L}(W_n),N(0,1))=\mathcal{O}(n^{-1/2})+\mathcal{O}(M_n^{1/p})$, with sharper rates up to $1/2$ when mixing decays sufficiently fast. The cumulants satisfy explicit bounds, enabling concentration and non-uniform Berry–Esseen bounds for the field, and the genogram-based approach provides a scalable combinatorial handle on the expansion terms. Together, these results yield sharp transport-distance bounds for dependent spatial data, matching i.i.d. rates under strong mixing and broadening the applicability of Stein-type methods to dependent structures.
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 $α$-mixing conditions. In this paper, we extend the Wasserstein-$p$ bounds to $α$-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 $O(|T|^{-β})$, where $|T|$ is the size of the index set, and $β\in (0, 1/2]$ depends on $p$, the dimension $d$ of the random fields, and the decay rate of the $α$-mixing coefficients. Notably, $β= 1/2$ 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.