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

2026/07/20 by Florian Heinrichs
#math.ST #stat.TH

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Abstract

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 L²([0,1]). Weak convergence is further derived for processes indexed by totally bounded subsets of L²([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.

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