Wasserstein-p Central Limit Theorem Rates: From Local Dependence to Markov Chains

• URL: http://arxiv.org/abs/2601.08184v2
• Date: Thu, 15 Jan 2026 03:46:02 GMT
• Title: Wasserstein-p Central Limit Theorem Rates: From Local Dependence to Markov Chains
• Abstract summary: We study CLT rates for multivariate dependent data in Wasserstein-$p$ ($W_p$) distance, for general $p geq 1$.<n>We focus on two fundamental dependence structures that commonly arise in machine learning: locally dependent sequences and geometrically ergodic Markov chains.
• Score: 19.40811084751516
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Finite-time central limit theorem (CLT) rates play a central role in modern machine learning. In this paper, we study CLT rates for multivariate dependent data in Wasserstein-$p$ ($W_p$) distance, for general $p \geq 1$. We focus on two fundamental dependence structures that commonly arise in machine learning: locally dependent sequences and geometrically ergodic Markov chains. In both settings, we establish the first optimal $O(n^{-1/2})$ rate in $W_1$, as well as the first $W_p$ ($p\ge 2$) CLT rates under mild moment assumptions, substantially improving the best previously known bounds in these dependent-data regimes. As an application of our optimal $W_1$ rate for locally dependent sequences, we further obtain the first optimal $W_1$-CLT rate for multivariate $U$-statistics. On the technical side, we derive a tractable auxiliary bound for $W_1$ Gaussian approximation errors that is well suited for studying dependent data. For Markov chains, we further prove that the regeneration time of the split chain associated with a geometrically ergodic chain has a geometric tail without assuming strong aperiodicity or other restrictive conditions. These tools may be of independent interests and enable our optimal $W_1$ rates and underpin our $W_p$ ($p\ge 2$) results.

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