Wasserstein-Based Test for Empirical Measure Convergence of Dependent Sequences

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 $T_n=\sqrt{n}{W}_1({\widehat{\mu }}_n,\mu )$ 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 $\sqrt{n}{W}_1({\widehat{\mu }}_n^{(i)},{\widehat{\mu }}_n^{(j)})$ 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.