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Wasserstein-Based Test for Empirical Measure Convergence of Dependent Sequences

Alexander Yordanov, Peter Hristov

Abstract

We develop Wasserstein-based hypothesis tests for empirical-measure convergence in stationary dependent sequences. For a known candidate invariant measure $μ$, we study the statistic $T_n=\sqrt{n}\,W_1(\hatμ_n,μ)$ and establish asymptotic level-$α$ 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 $\sqrt{n}\,W_1(\hatμ_n^{(i)},\hatμ_n^{(j)})$ for independent trajectories and obtain a corresponding pairwise test, including Bonferroni control for multiple comparisons. Simulation experiments involving both linear and nonlinear dynamical settings illustrate both the coverage probability and the power of the tests.

Wasserstein-Based Test for Empirical Measure Convergence of Dependent Sequences

Abstract

We develop Wasserstein-based hypothesis tests for empirical-measure convergence in stationary dependent sequences. For a known candidate invariant measure , we study the statistic and establish asymptotic level- 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 for independent trajectories and obtain a corresponding pairwise test, including Bonferroni control for multiple comparisons. Simulation experiments involving both linear and nonlinear dynamical settings illustrate both the coverage probability and the power of the tests.

Paper Structure

This paper contains 8 sections, 5 theorems, 70 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Testing Empirical Measure Convergence to a Known Invariant Measure
  3. Pairwise Convergence Testing with Two Empirical Measures
  4. Simulation Results
  5. Known Finite Covariance
  6. Known Infinite Covariance
  7. Estimated Infinite Covariance
  8. Conclusion

Key Result

Proposition 1

Under the same assumptions as those in Proposition 3.1 in Dedecker:2017 and assuming that $\mathbb P(\mathbb G=q_{1-\alpha})=0$. Then the test in Definition def:one-sample-test has asymptotic level $\alpha$, i.e., equivalently, $\blacktriangleleft$$\blacktriangleleft$

Figures (5)

  • Figure 1: Comparison of the distribution of the scaled empirical Wasserstein statistics $\sqrt{n}\,W_1\bigl(\hat{\mu}_n^{(i)},\hat{\mu}_n^{(j)}\bigr)$ and the theoretical limiting distribution for an $\operatorname{MA}(3)$ sequence. (a) Null case: trajectories share one invariant distribution; (b) Alternative case: trajectories come from two different invariant distributions.
  • Figure 2: A kernel-density estimate of the sampling distributions of $\sqrt{n}\,W_1\!\left(\hat{\mu}_n^{(i)},\hat{\mu}_n^{(j)}\right)$ under the alternative in an $\operatorname{MA}(3)$ setting, showing rightward drift as $n$ increases.
  • Figure 3: Comparison of the distribution of the scaled empirical Wasserstein statistics $\sqrt{n}\,W_1\bigl(\hat{\mu}_n^{(i)},\hat{\mu}_n^{(j)}\bigr)$ and the theoretical limiting distribution for an $\operatorname{ARMA}(5,3)$ sequence in the convergent case where the Wasserstein statistic agrees well with the asymptotic limit distribution under $H_0$.
  • Figure 4: Double-pendulum geometry used in the long-run covariance estimation experiment.
  • Figure 5: Comparison of the distribution of the scaled empirical Wasserstein statistics $\sqrt{n}\,W_1\bigl(\hat{\mu}_n^{(i)},\hat{\mu}_n^{(j)}\bigr)$ and the theoretical limiting distribution for the four observables of a double pendulum. (a) Convergent setting: trajectories from the same invariant distribution; (b) Divergent setting: trajectories from different invariant distributions.

Theorems & Definitions (13)

  • Definition 1: Wasserstein test for one invariant measure
  • Proposition 1: Asymptotic coverage under $H_0$
  • proof
  • Proposition 2: Power under fixed alternatives
  • proof
  • Theorem 3: Asymptotic distribution of the two-sample Wasserstein distance
  • proof
  • Definition 2: Asymptotic pairwise Wasserstein test
  • Proposition 4: Asymptotic coverage under $H_0$
  • proof
  • ...and 3 more