Wasserstein Concentration of Empirical Measures for Dependent Data via the Method of Moments

TL;DR

This paper introduces a novel concentration inequality for the 1-Wasserstein distance between empirical and true measures in dependent data, relying on moment control and tail conditions rather than independence.

Contribution

It provides a general concentration result applicable to dependent sequences using only variance and tail conditions, expanding beyond traditional independence assumptions.

Findings

Establishes Wasserstein concentration under dependence with moment and tail controls

Uses polynomial approximation to connect moment bounds to measure concentration

Applicable to data with significant dependencies, broadening existing theories

Abstract

We establish a general concentration result for the 1-Wasserstein distance between the empirical measure of a sequence of random variables and its expectation. Unlike standard results that rely on independence (e.g., Sanov's theorem) or specific mixing conditions, our result requires only two conditions: (1) control over the variance of the empirical moments, and (2) a flexible tail condition we term $\Psi_{r_n}$-sub-Gaussianity. This approach allows for significant dependencies between variables, provided their algebraic moments behave predictably. The proof uses the method of moments combined with a polynomial approximation of Lipschitz functions via Jackson kernels, allowing us to translate moment concentration into topological concentration.