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

Abstract

We establish normal-approximation bounds in the Wasserstein-$p$ distance for $\alpha$-mixing random fields, extending results previously available mainly for independent or locally dependent variables. For standardized sums indexed by $T\subset {\mathbb{Z}}^d$, we obtain rates $O(|T{|}^{-\beta })$, where $\beta \in (0,1/2]$ depends on $p$, the moment order, the dimension, and the mixing decay; the optimal square-root rate holds under sufficiently fast polynomial mixing. For integer $p\ge 2$, a refined bound retains the stronger mixing power produced by the expansion, covers the endpoint of finite $(p+2)$-th moments, and identifies exact logarithmic corrections at critical decay. Under finite-range dependence, the endpoint moment condition gives the optimal rate for every real $p\ge 1$. The proof combines a cumulant-based Edgeworth expansion with Stein's method. Its main technical tool is a constructive graph calculus that organizes high-order dependence and preserves cancellations before mixing bounds are applied. The resulting Wasserstein estimates also imply two-regime non-uniform normal-approximation bounds with sharper sample-size decay in the far tail. A Lean 4/Mathlib companion machine-checks the principal theorem interfaces and the complete paper-internal genogram argument for the expansions, relative to seven explicitly identified literature and standard inputs.