Bounds in Wasserstein distance for locally stationary functional time series

We investigate nonparametric estimation of conditional distributions for locally stationary functional time series (LSFTS) within a general semi-metric framework. The response variable is scalar, while the covariate process evolves in an infinite-dimensional space and exhibits smooth temporal nonstationarity. We propose a Nadaraya–Watson-type estimator formulated as a locally weighted empirical measure that jointly localizes in rescaled time and functional covariate space. Under mild structural assumptions encompassing small-ball probability conditions, weak $$\beta $$ -mixing dependence, and minimal smoothness of the conditional distribution function, we establish explicit convergence rates for the estimator in Wasserstein distance. The analysis reveals how temporal localization, functional complexity, and dependence interact to govern distributional convergence in nonstationary functional settings. Extensions to higher-order Wasserstein metrics are derived under boundedness conditions. Comprehensive numerical experiments, including locally stationary functional autoregressive models and real-world datasets, corroborate the theoretical findings and illustrate the practical effectiveness of the proposed methodology for distributional inference beyond conditional means.