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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.

Submission history

From: Tianle Liu [view email]
[v1] Wed, 13 Sep 2023 15:36:44 UTC (109 KB)
[v2] Wed, 6 Dec 2023 03:44:35 UTC (112 KB)
[v3] Sun, 22 Sep 2024 15:38:38 UTC (132 KB)
[v4] Sun, 23 Feb 2025 19:39:35 UTC (125 KB)
[v5] Sun, 30 Aug 2026 04:09:05 UTC (112 KB)