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

Arash A. Amini, Luciano Vinas

TL;DR

The paper proves a general $W_1$-concentration result for the empirical measure of dependent data by only requiring asymptotic control of empirical moments and a flexible $\Psi_{r_n}$-sub-Gaussian tail condition. It leverages the method of moments and a Chebyshev–Jackson polynomial approximation to transfer moment concentration into Lipschitz-test concentration, yielding a robust convergence result beyond i.i.d. or strongly mixing scenarios. The authors extend the scalar result to random vectors via a projection-based lifting argument, enabling high-dimensional applicability. The approach provides a practical framework for concentration in settings with complex dependencies, such as random matrices, particle systems, or graph-based models, and introduces explicit polynomial-approximation tools with controlled coefficients to bridge moment and topological concentration.

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 $Ψ_{r_n}$-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.

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

TL;DR

The paper proves a general -concentration result for the empirical measure of dependent data by only requiring asymptotic control of empirical moments and a flexible -sub-Gaussian tail condition. It leverages the method of moments and a Chebyshev–Jackson polynomial approximation to transfer moment concentration into Lipschitz-test concentration, yielding a robust convergence result beyond i.i.d. or strongly mixing scenarios. The authors extend the scalar result to random vectors via a projection-based lifting argument, enabling high-dimensional applicability. The approach provides a practical framework for concentration in settings with complex dependencies, such as random matrices, particle systems, or graph-based models, and introduces explicit polynomial-approximation tools with controlled coefficients to bridge moment and topological concentration.

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 -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.
Paper Structure (8 sections, 7 theorems, 42 equations)

This paper contains 8 sections, 7 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The $\Psi_{r}$ Norm and Asymptotic Sub-Gaussianity
  4. Main Results
  5. Extension to Random Vectors
  6. Polynomial approximations
  7. Proof of Main Result (Scalar Case)
  8. Proof of Vector Extension (Theorem \ref{['thm:vector_result']})

Key Result

Lemma 2

Let $r \ge 2$ and $\|X\|_{\Psi_r} \le K$. Then:

Theorems & Definitions (14)

  • Definition 1: Truncated $\Psi_r$ Norm
  • Lemma 2: Moments and Tails
  • Theorem 3
  • Theorem 4
  • Lemma 5
  • proof
  • Lemma 6: Chebyshev--Jackson approximation
  • proof
  • proof : Proof of Theorem \ref{['thm:main_result']}
  • Lemma 7: Lipschitz Continuity of Projections
  • ...and 4 more