The adapted Wasserstein distance controls the calibration errors of optimal values in various stochastic optimization problems, pricing and hedging problems, optimal stopping problems, etc. Motivated by approximating the true underlying distribution by empirical data, we consider empirical measures of $\mathbb{R}^d$-valued stochastic process in finite discrete-time. It is known that the empirical measures do not converge under the adapted Wasserstein distance. To address this issue, we consider convolutions of Gaussian kernels and empirical measures as an alternative, which we refer to the Gaussian-smoothed empirical measures. By setting the bandwidths of Gaussian kernels depending on the number of samples, we prove the convergence of the Gaussian-smoothed empirical measures to the true underlying measure in terms of mean, deviation, and almost sure convergence. Although Gaussian-smoothed empirical measures converge to the true underlying measure and can potentially enlarge data, they are not discrete measures and therefore not applicable in practice. Therefore, we combine Gaussian-smoothed empirical measures and the adapted empirical measures in \cite{acciaio2022convergence} to introduce the adapted smoothed empirical measures, which are discrete substitutes of the smoothed empirical measures. We establish the polynomial mean convergence rate, the exponential deviation convergence rate and the almost sure convergence of the adapted smoothed empirical measures.

翻译：适应化Wasserstein距离控制了各类随机优化问题、定价与对冲问题、最优停止问题等中优化值的校准误差。受通过经验数据近似真实潜在分布的启发，我们考虑有限离散时间下$\mathbb{R}^d$值随机过程的经验测度。已知经验测度在适应化Wasserstein距离下不收敛。为解决该问题，我们考虑高斯核与经验测度的卷积作为替代方案，并将其称为高斯平滑经验测度。通过设定依赖于样本数量的高斯核带宽，我们证明了高斯平滑经验测度在均值、偏差及几乎必然收敛意义下收敛至真实潜在分布。尽管高斯平滑经验测度能收敛至真实潜在分布并可能实现数据扩增，但由于其非离散测度的特性难以应用于实践。为此，我们将高斯平滑经验测度与文献\cite{acciaio2022convergence}中的适应化经验测度相结合，引入适应化平滑经验测度作为平滑经验测度的离散替代。我们建立了适应化平滑经验测度的多项式均值收敛速度、指数偏差收敛速度及几乎必然收敛性。