PreprintCC BY 4.0arXiv
Can we further improve the central limit theorem for stationary ρ-mixing sequences ?
Abstract
Through the seminal works of Peligrad and Bradley, the existing sufficient conditions for the central limit theorem for ρ-mixing sequences are known to be very close to necessary. Nevertheless, we show that further improvements to these conditions are achievable when the ρ-mixing coefficients are non-summable, bridging a long-standing gap in the borderline case. The key idea is an iterative block-size technique that enables precise evaluations of the slowly varying variance components. Finally, known examples and new extensions--including iterated logarithmic mixing rates--are investigated to demonstrate the sharpness of our criteria, revealing a clear parameter structure and posing related open problems.
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BibTeX
@misc{matsui2026further,
title = {{Can we further improve the central limit theorem for stationary $ρ$-mixing sequences $?$}},
author = {Muneya Matsui},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04638},
eprint = {2609.04638},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04638},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Can we further improve the central limit theorem for stationary ρ-mixing sequences ?
AU - Muneya Matsui
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.04638
UR - https://arxiv.org/abs/2609.04638
AB - Through the seminal works of Peligrad and Bradley, the existing sufficient conditions for the central limit theorem for ρ-mixing sequences are known to be very close to necessary. Nevertheless, we show that further improvements to these conditions are achievable when the ρ-mixing coefficients are non-summable, bridging a long-standing gap in the borderline case. The key idea is an iterative block-size technique that enables precise evaluations of the slowly varying variance components. Finally, known examples and new extensions--including iterated logarithmic mixing rates--are investigated to demonstrate the sharpness of our criteria, revealing a clear parameter structure and posing related open problems.
ER -