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

TL;DR

This paper establishes optimal non-asymptotic Wasserstein-$p$ CLT rates for dependent data, including locally dependent sequences and Markov chains, improving previous bounds and enabling new applications.

Contribution

It provides the first optimal $W_1$ CLT rate for locally dependent sequences and multivariate U-statistics, and introduces new technical tools for dependent data analysis.

Findings

Achieved $ ext{O}(n^{-1/2})$ rate in $W_1$ for dependent sequences.

Established $W_p$ CLT rates for $p eq 1$ under mild moments.

Proved geometric tail bounds for Markov chain regeneration times.

Abstract

Non-asymptotic central limit theorem (CLT) rates play a central role in modern machine learning and operations research. In this paper, we study CLT rates for multivariate dependent data in Wasserstein-$p$ ($W_p$) distance, for general $p\ge 1$. We focus on two fundamental dependence structures that commonly arise in practice: locally dependent sequences and geometrically ergodic Markov chains. In both settings, we establish the first optimal $\mathcal O(n^{-1/2})$ 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.