标题： Wasserstein距离中局部平稳过程的界 显示英文标题

标题： Bounds in Wasserstein Distance for Locally Stationary Processes

摘要： 局部平稳（LSPs）构成了捕捉时间序列数据中固有细微动态的重要建模范式，这些数据的统计特征，包括均值和方差，在时间上平滑地演变。 在本文中，我们引入了一种专门针对LSPs的新型条件概率分布估计器，采用Nadaraya-Watson（NW）核平滑方法。 NW估计器是一种重要的局部平均技术，利用核平滑来近似给定协变量的响应变量的条件分布。 我们在Wasserstein度量下严格建立了基于NW的单变量条件概率估计器的收敛速率，提供了明确的界限和保证最优性能的条件。 扩展这一理论框架，我们随后使用切片Wasserstein距离将分析推广到多变量情况，这种方法在克服高维设置通常关联的计算和分析挑战方面尤其有利。 为了验证我们的理论贡献，我们在合成数据集上进行了广泛的数值模拟，并使用真实世界数据提供了实证验证，突出了该估计器在捕捉复杂时间依赖性方面的实际相关性和有效性，并强调了其在分析复杂非平稳现象中的相关性。 显示英文摘要

摘要： Locally stationary (LSPs) constitute an essential modeling paradigm for capturing the nuanced dynamics inherent in time series data whose statistical characteristics, including mean and variance, evolve smoothly across time. In this paper, we introduce a novel conditional probability distribution estimator specifically tailored for LSPs, employing the Nadaraya-Watson (NW) kernel smoothing methodology. The NW estimator, a prominent local averaging technique, leverages kernel smoothing to approximate the conditional distribution of a response variable given its covariates. We rigorously establish convergence rates for the NW-based conditional probability estimator in the univariate setting under the Wasserstein metric, providing explicit bounds and conditions that guarantee optimal performance. Extending this theoretical framework, we subsequently generalize our analysis to the multivariate scenario using the sliced Wasserstein distance, an approach particularly advantageous in circumventing the computational and analytical challenges typically associated with high-dimensional settings. To corroborate our theoretical contributions, we conduct extensive numerical simulations on synthetic datasets and provide empirical validations using real-world data, highlighting the estimator's practical relevance and effectiveness in capturing intricate temporal dependencies and underscoring its relevance for analyzing complex nonstationary phenomena.