Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2309.07031 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Table of Contents:
- 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.