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. 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.

翻译：针对平稳相依序列中的经验测度收敛问题，我们发展了基于Wasserstein距离的假设检验方法。对于已知候选不变测度$μ$，我们考察统计量$T_n=\sqrt{n}\,W_1(\hatμ_n,μ)$，并在原假设下建立渐近水平$α$有效性，同时在固定备择假设下证明其一致性。当不变测度未知时，我们推导了独立轨迹对统计量$\sqrt{n}\,W_1(\hatμ_n^{(i)},\hatμ_n^{(j)})$的渐近分布，并建立相应的成对检验方法，包括多重比较的Bonferroni校正。为在长期协方差无法显式获得时实现可行性估计，我们引入有限网格插件估计量，并证明基于估计协方差的高斯临界值可一致地恢复相应的先知固定网格估计。在线性与非线性动力学场景下的仿真实验分别展示了先知方法和插件方法的表现，及其覆盖概率与检验功效。