Wasserstein-p Bounds via Cumulant-Based Edgeworth Expansion for $α$-Mixing Random 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.