Wasserstein Kernel Regression with Dependent Distribution-Valued Predictors: Local Recurrence, Joint Inference, and Multivariate Gaussian Geometry

We study scalar nonparametric regression with a probability-measure-valued predictor observed along a dependent time series. A Nadaraya–Watson smoother localizes directly in the quadratic Wasserstein metric. The revised theory separates generic functional-kernel arguments from contributions generated by Wasserstein predictor geometry and local recurrence. We introduce an aggregate local-recurrence condition that is weaker than pairwise anti-clustering and is compatible with strongly overlapping empirical-distribution windows. For one-dimensional Gaussian location–scale predictors, we obtain an exact Euclidean small-ball expansion; we then extend the geometry to nonsingular multivariate Gaussian predictors through the Bures–Wasserstein Riemannian metric, obtaining an explicit local dimension and small-ball exponent. Under geometric strong mixing, local Hölder smoothness, and small-ball regularity, we establish strong consistency, an almost-sure rate, a variance expansion, and a triangular-array central limit theorem. We further derive a local long-run-variance CLT, a finite-dimensional joint CLT for several Wasserstein targets, and simultaneous confidence intervals. Feasible inference uses a direct Bartlett–HAC estimator whose consistency is now justified by an explicit fourth-order local cumulant condition rather than by geometric mixing alone. For tuning, we introduce an empirical small-ball admissibility screen and allow selection from a possibly growing grid on a separated chronological block. A truncation extension handles locally unbounded responses. Simulations satisfy the stated bounded-response assumptions and stress-test overlapping windows. The Mauna Loa CO2 illustration adds interpolation, window/horizon, and deterministic-overlap sensitivity, stationarity diagnostics, coefficient intervals, and a stronger AIC-selected autoregressive comparator. The latter can outperform the Wasserstein smoother, so the application is interpreted as an empirical illustration rather than evidence of universal predictive superiority.