Time-Uniform Dependent Central Limit Theory
Abstract
We establish a near-optimal almost sure invariance principle for nonstationary time series under the functional dependence measure (FDM) framework. This result yields a universal time-uniform dependent central limit theorem, which serves as the basis for anytime-valid inference. In particular, we instantiate the theorem through asymptotic confidence sequences (AsympCSs), time-uniform counterparts of asymptotic confidence intervals obtained via central limit theorems, extending those originally proposed in Waudby-Smith et al. (2024) to time series with serial dependence. We further refine their definition because, as we show, it can admit AsympCSs that fail to achieve the desired asymptotic time-uniform coverage for a broad class of data-generating processes, including independent and identically distributed cases. We address this issue by adding a simple eventual-inclusion condition, which in turn permits more flexible variance estimation, particularly useful in time-series settings. For feasible implementation of AsympCSs, we establish strong consistency of HAC variance estimators under the FDM framework, accommodating a wider range of commonly used bandwidth choices than existing results under mixing conditions. As illustrations, we construct AsympCSs for algorithms with Markovian structure and time-series-based causal estimands.
Related
- Central limit theory for serial tail dependence estimators in heavy-tailed long memory linear time series
- Central limit theorems for high-dimensional ridge regression under uniform subsampling
- The semiclassical limit from Hartree to Vlasov at positive density: strong uniform-in-time convergence and scattering
- The uniform-in-time electrostatic limit of the linearised Vlasov-Maxwell system: the Poisson equilibrium
- Response Time Central-Limit and Failure Rate Estimation for Stationary Periodic Rate Monotonic Real-Time Systems
BibTeX
@misc{ge2026timeuniform,
title = {{Time-Uniform Dependent Central Limit Theory}},
author = {Mingxuan Ge and Yanlong Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29288},
eprint = {2610.29288},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29288},
}
RIS
TY - UNPB
TI - Time-Uniform Dependent Central Limit Theory
AU - Mingxuan Ge
AU - Yanlong Liu
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29288
UR - https://arxiv.org/abs/2610.29288
AB - We establish a near-optimal almost sure invariance principle for nonstationary time series under the functional dependence measure (FDM) framework. This result yields a universal time-uniform dependent central limit theorem, which serves as the basis for anytime-valid inference. In particular, we instantiate the theorem through asymptotic confidence sequences (AsympCSs), time-uniform counterparts of asymptotic confidence intervals obtained via central limit theorems, extending those originally proposed in Waudby-Smith et al. (2024) to time series with serial dependence. We further refine their definition because, as we show, it can admit AsympCSs that fail to achieve the desired asymptotic time-uniform coverage for a broad class of data-generating processes, including independent and identically distributed cases. We address this issue by adding a simple eventual-inclusion condition, which in turn permits more flexible variance estimation, particularly useful in time-series settings. For feasible implementation of AsympCSs, we establish strong consistency of HAC variance estimators under the FDM framework, accommodating a wider range of commonly used bandwidth choices than existing results under mixing conditions. As illustrations, we construct AsympCSs for algorithms with Markovian structure and time-series-based causal estimands.
ER -