A Functional Central Limit Theorem for Locally Stationary Time Series in Banach Spaces
Abstract
We establish a functional central limit theorem for locally stationary time series in a separable Banach space B. The result requires no type-2 or cotype assumptions and thus covers spaces central to functional data analysis, such as C([0,1]) and Lp([0,1]). Under moment conditions, summable physical dependence coefficients and a bracketing entropy condition controlling the infinite-dimensional tails, the centered partial sum process converges weakly in $D([0,1],B)$ to a centered Gaussian process, whose covariance is the integrated local long-run covariance, an element of the projective tensor product. We also obtain a stochastic integral representation with respect to a cylindrical Brownian motion. As the limit depends on the unknown long-run covariance, we prove consistency of a block multiplier bootstrap for the partial sum process. This yields CUSUM tests for a constant mean of the full process, based on arbitrary continuous seminorms and valid under a time-varying long-run covariance. For linear projections, we derive a self-normalized test with a pivotal limit. In simulations, the supremum norm test detects localized changes that its L2 counterpart largely misses, and applications to EEG and temperature data illustrate the methodology. Examples in C([0,1]) and W1,p0([0,1]) show how the entropy condition can be verified.
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BibTeX
@misc{heinrichs2026functional,
title = {{A Functional Central Limit Theorem for Locally Stationary Time Series in Banach Spaces}},
author = {Florian Heinrichs and Luis-Alberto Rodríguez},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.25791},
eprint = {2609.25791},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25791},
}
RIS
TY - UNPB
TI - A Functional Central Limit Theorem for Locally Stationary Time Series in Banach Spaces
AU - Florian Heinrichs
AU - Luis-Alberto Rodríguez
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2609.25791
UR - https://arxiv.org/abs/2609.25791
AB - We establish a functional central limit theorem for locally stationary time series in a separable Banach space B. The result requires no type-2 or cotype assumptions and thus covers spaces central to functional data analysis, such as C([0,1]) and L^p([0,1]). Under moment conditions, summable physical dependence coefficients and a bracketing entropy condition controlling the infinite-dimensional tails, the centered partial sum process converges weakly in $D([0,1],B)$ to a centered Gaussian process, whose covariance is the integrated local long-run covariance, an element of the projective tensor product. We also obtain a stochastic integral representation with respect to a cylindrical Brownian motion. As the limit depends on the unknown long-run covariance, we prove consistency of a block multiplier bootstrap for the partial sum process. This yields CUSUM tests for a constant mean of the full process, based on arbitrary continuous seminorms and valid under a time-varying long-run covariance. For linear projections, we derive a self-normalized test with a pivotal limit. In simulations, the supremum norm test detects localized changes that its L^2 counterpart largely misses, and applications to EEG and temperature data illustrate the methodology. Examples in C([0,1]) and W^1,p_0([0,1]) show how the entropy condition can be verified.
ER -