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arxiv: 2604.02700 · v2 · pith:2DKHTONWnew · submitted 2026-04-03 · 📊 stat.AP

Wasserstein-Based Test for Empirical Measure Convergence of Dependent Sequences

Alexander Yordanov , Peter Hristov This is my paper

Pith reviewed 2026-05-22 10:09 UTC · model grok-4.3

classification 📊 stat.AP
keywords Wasserstein distanceempirical measuresdependent sequenceshypothesis testinginvariant measurescentral limit theoremstationary processes
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The pith

Wasserstein-1 distance yields asymptotically valid tests for empirical measure convergence in stationary dependent sequences.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs hypothesis tests that use the Wasserstein-1 distance to assess whether the empirical measure of a stationary dependent sequence converges to a candidate invariant measure. For a known target measure the scaled distance statistic admits a limiting distribution that delivers level-alpha tests under the null together with consistency against fixed alternatives. When the target is unknown, pairwise distances computed on independent trajectories follow a Gaussian limit after scaling, and Bonferroni adjustment produces valid multiple-comparison procedures. A plug-in estimator for the long-run covariance allows the tests to be implemented without closed-form expressions for the covariance operator.

Core claim

For a known invariant measure mu the statistic sqrt(n) W_1(hat mu_n, mu) has an asymptotic distribution that permits construction of level-alpha tests under the null of convergence and is consistent against fixed alternatives. When mu is unknown the pairwise statistic sqrt(n) W_1(hat mu_n^(i), hat mu_n^(j)) converges to a centered Gaussian random variable, supporting Bonferroni-controlled pairwise tests. A finite-grid plug-in estimator recovers the same limiting law as the oracle covariance and is shown to yield consistent critical values.

What carries the argument

The Wasserstein-1 distance W_1 applied to the empirical measure process, whose limiting distribution is obtained from the long-run covariance operator of the stationary sequence in the space of probability measures.

If this is right

  • The test maintains correct asymptotic size for any stationary sequence whose long-run covariance is positive definite.
  • Power approaches one against any fixed deviation of the empirical measure from the candidate invariant measure.
  • Pairwise comparisons with Bonferroni correction control the family-wise error rate when the invariant measure must be estimated from the data.
  • The finite-grid plug-in covariance estimator produces tests whose critical values converge to those of the oracle procedure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method supplies a practical diagnostic for whether numerical trajectories of a dynamical system have reached their theoretical invariant distribution.
  • Extensions to other Wasserstein orders or to non-stationary sequences would require only local modifications of the covariance operator.
  • In high-dimensional state spaces the computational cost of W_1 can be reduced by entropic regularization without altering the asymptotic justification.

Load-bearing premise

The observed sequences are stationary and the long-run covariance operator of the empirical-measure process exists and is positive definite.

What would settle it

Large-sample Monte Carlo experiments under the null that generate data from a stationary process satisfying the covariance condition yet produce rejection rates that fail to approach the nominal alpha level would falsify the claimed asymptotic validity.

Figures

Figures reproduced from arXiv: 2604.02700 by Alexander Yordanov, Peter Hristov.

Figure 1
Figure 1. Figure 1: Comparison of the distribution of the scaled empirical Wasserstein statistics [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A kernel-density estimate of the sampling distributions of [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of the distribution of the scaled empirical Wasserstein statistics [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Double-pendulum geometry used in the long-run covariance estimation experiment. [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of the distribution of the scaled empirical Wasserstein statistics [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
read the original abstract

We develop Wasserstein-based hypothesis tests for empirical-measure convergence in stationary dependent sequences. For a known candidate invariant measure, $\mu$, we study the statistic $T_n=\sqrt{n}\,W_1(\hat\mu_n,\mu)$ and establish asymptotic level-$\alpha$ validity under the null, together with consistency under fixed alternatives. When the invariant measure is unknown, we derive the asymptotic law of the pairwise statistic $\sqrt{n}\,W_1(\hat\mu_n^{(i)},\hat\mu_n^{(j)})$ for independent trajectories and obtain a corresponding pairwise test, including Bonferroni control for multiple comparisons. To make this estimation feasible when the long-run covariance is unavailable in closed form, we introduce a finite-grid plug-in estimator and show that Gaussian critical values based on the estimated covariance consistently recover the corresponding oracle fixed-grid estimation. Simulation experiments in both linear and nonlinear dynamical settings illustrate the oracle and plug-in regimes, along with the resulting coverage probability and power.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops Wasserstein-based hypothesis tests for empirical-measure convergence in stationary dependent sequences. For a known invariant measure μ, the statistic T_n = sqrt(n) W_1(μ̂_n, μ) is shown to have asymptotic level-α validity under the null and consistency under fixed alternatives. For unknown μ, the asymptotic law of the pairwise statistic sqrt(n) W_1(μ̂_n^(i), μ̂_n^(j)) between independent trajectories is derived, yielding a Bonferroni-controlled pairwise test. A finite-grid plug-in estimator for the long-run covariance is introduced, with the claim that Gaussian critical values from the estimated covariance recover the oracle fixed-grid behavior. Simulations in linear and nonlinear dynamical systems illustrate coverage probabilities and power.

Significance. If the derivations hold, the paper offers a practical framework for testing convergence to invariant measures via Wasserstein distances in dependent data, relevant to time series and dynamical systems. The finite-grid plug-in addresses cases where closed-form long-run covariances are unavailable, and the simulations provide empirical checks. The work extends empirical-process ideas to Wasserstein space and could be useful if the limiting laws and critical-value approximations are rigorously justified.

major comments (2)
  1. [Abstract and pairwise-statistic derivation] Abstract and derivation of the pairwise statistic: The claim that sqrt(n) W_1(μ̂_n^(i), μ̂_n^(j)) admits a Gaussian limit (enabling direct use of Gaussian critical values from the plug-in covariance) appears inconsistent with standard continuous-mapping arguments. Under the assumed CLT sqrt(n)(μ̂_n - μ) ⇒ G (Gaussian random measure), the limit of the pairwise statistic is the law of sup_{Lip(f)≤1} |∫ f d(G^i - G^j)|, the supremum norm of a centered Gaussian process indexed by the unit Lipschitz ball. This limit is non-Gaussian in general. The manuscript must clarify in which section or theorem the asymptotic law is derived and how Gaussianity (or a valid Gaussian approximation) is obtained for the pairwise case.
  2. [Finite-grid plug-in estimator] Finite-grid plug-in estimator section: The statement that Gaussian critical values based on the estimated covariance consistently recover the oracle fixed-grid estimation is load-bearing for the practical test. If the Wasserstein distance remains a supremum functional even after gridding, the limiting distribution is still that of a sup of Gaussians rather than a Gaussian random variable. Please specify the grid construction, the precise form of the plug-in covariance estimator, and the justification (e.g., linearization, delta-method, or explicit simulation of the sup) that validates the Gaussian critical values.
minor comments (2)
  1. [Abstract / Introduction] The abstract invokes the existence of a positive-definite long-run covariance operator but does not state the precise function space or topology in which the CLT for the empirical measure holds; this should be made explicit early in the manuscript.
  2. [Simulation section] Simulation experiments would benefit from reporting the specific grid sizes used and sensitivity checks on coverage as grid resolution varies.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comments highlight important points about the limiting distribution of the pairwise statistic and the justification for the finite-grid approximation. We address each major comment below and indicate the revisions we will make to improve clarity and rigor.

read point-by-point responses

  1. Referee: [Abstract and pairwise-statistic derivation] Abstract and derivation of the pairwise statistic: The claim that sqrt(n) W_1(μ̂_n^(i), μ̂_n^(j)) admits a Gaussian limit (enabling direct use of Gaussian critical values from the plug-in covariance) appears inconsistent with standard continuous-mapping arguments. Under the assumed CLT sqrt(n)(μ̂_n - μ) ⇒ G (Gaussian random measure), the limit of the pairwise statistic is the law of sup_{Lip(f)≤1} |∫ f d(G^i - G^j)|, the supremum norm of a centered Gaussian process indexed by the unit Lipschitz ball. This limit is non-Gaussian in general. The manuscript must clarify in which section or theorem the asymptotic law is derived and how Gaussianity (or a valid Gaussian approximation) is obtained for the pairwise case.

    Authors: We appreciate this clarification. Theorem 3.2 derives the asymptotic law of the pairwise statistic precisely as the supremum of the absolute value of the centered Gaussian process indexed by the unit Lipschitz functions, which is non-Gaussian. The abstract and introduction refer to this limiting law and then separately introduce the finite-grid plug-in to obtain practical critical values. We will revise the abstract to explicitly state that the limiting distribution is that of the supremum functional and that Gaussian critical values are used as an approximation for the fixed-grid version of the statistic. A new remark will be added after Theorem 3.2 explaining that the Gaussian approximation is obtained by discretizing to a finite-dimensional vector whose covariance is estimated by the plug-in, with the approximation error controlled as the grid is refined. revision: partial

  2. Referee: [Finite-grid plug-in estimator] Finite-grid plug-in estimator section: The statement that Gaussian critical values based on the estimated covariance consistently recover the oracle fixed-grid estimation is load-bearing for the practical test. If the Wasserstein distance remains a supremum functional even after gridding, the limiting distribution is still that of a sup of Gaussians rather than a Gaussian random variable. Please specify the grid construction, the precise form of the plug-in covariance estimator, and the justification (e.g., linearization, delta-method, or explicit simulation of the sup) that validates the Gaussian critical values.

    Authors: We agree that the current presentation is too terse. The grid is a fixed finite partition of the compact metric space into M cells, with the empirical measure replaced by its vector of cell probabilities. On this grid the Wasserstein-1 distance reduces to a weighted L1 norm of the probability vectors (with weights given by the cell diameters). The plug-in covariance estimator is the sample long-run covariance matrix of the M-dimensional time series of these probability vectors, computed with a Bartlett kernel whose bandwidth is chosen by the Andrews automatic procedure. Because the statistic is now a continuous function of a finite-dimensional Gaussian vector, the delta method yields asymptotic normality, and the Gaussian critical values are the quantiles of this limiting normal distribution. We will add an expanded subsection with the explicit formulas for the grid projection, the kernel estimator, the consistency proof for the critical values, and a bound on the discretization error relative to the continuous-space supremum. revision: yes

Circularity Check

0 steps flagged

No circularity: asymptotics derived from standard empirical-process CLT

full rationale

The paper invokes the CLT for the empirical measure in Wasserstein space under stationarity and existence of a positive-definite long-run covariance operator, then applies continuous mapping to obtain the limiting law of sqrt(n) W_1(hat mu_n, mu) and the pairwise version. These steps rest on external empirical-process results rather than defining the target statistic or its critical values in terms of fitted parameters from the same data. The finite-grid plug-in is justified by approximation to an oracle estimator, not by self-definition. No self-citations, ansatzes, or uniqueness theorems from the authors' prior work appear as load-bearing steps. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on stationarity of the sequence, existence of a positive-definite long-run covariance operator in the Wasserstein space, and standard empirical-process CLTs; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The sequence is strictly stationary with a long-run covariance operator that exists and is positive definite.
    Required for the asymptotic distribution of sqrt(n) W_1(μ̂_n, μ) to be well-defined and for the plug-in estimator to be consistent.

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Works this paper leans on

5 extracted references · 5 canonical work pages

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    Behavior of the wasserstein distance between the empirical and the marginal distributions of stationaryα-dependent sequences.Bernoulli, 2017

    Jérôme Dedecker and Florence Merlevède. Behavior of the wasserstein distance between the empirical and the marginal distributions of stationaryα-dependent sequences.Bernoulli, 2017

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    Convergence of the empirical measure in expected wasserstein distance: non-asymptotic explicit bounds inRd.ESAIM: PS, 27:749–775, 2023

    Nicolas Fournier. Convergence of the empirical measure in expected wasserstein distance: non-asymptotic explicit bounds inRd.ESAIM: PS, 27:749–775, 2023

    work page 2023

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    Panaretos and Yoav Zemel

    Victor M. Panaretos and Yoav Zemel. Statistical aspects of wasserstein distances.Annual Review of Statistics and Its Application, 6:405–431, 2019

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    Springer, 2008

    Cédric Villani et al.Optimal Transport: Old and New, volume 338. Springer, 2008. 9 Appendix A. Double Pendulum Simulation Initial Conditions Sampling Algorithm As discussed in [1], it is appropriate to talk about ergodicity only for ensembles of Hamiltonian (i.e., measure-preserving) systems of the same total energy. Hence, given an initial energyE, we wa...

    work page 2008