A Functional Central Limit Theorem for Localized Partial Sums of Non-Stationary Time Series

A localized functional central limit theorem is established for kernel-weighted partial sum processes of piecewise locally stationary time series under geometric decay of the physical dependence measure. The localized process is shown to converge weakly to a centered Gaussian random distribution in $D'(0,1)$, and the limit extends naturally to an isonormal Gaussian process on $L^2([0,1])$. Weak convergence is further derived for processes indexed by totally bounded subsets of $L^2([0,1])$. As an application, the localized limit theory is used to construct tests for constant mean functions against linear, polynomial, and general alternatives in non-parametric regression with locally stationary errors. Simulation results and data examples illustrate the finite sample performance and practical applicability of the proposed methodology.