Invariance principle for long-memory linear processes under sublinear expectations
Abstract
We establish an invariance principle for long-memory linear processes with power-law coefficients under sublinear expectations. Under truncated-moment and capacity-tail conditions, we construct these processes on a canonical space and prove weak convergence in continuous path space of their linearly interpolated, normalized partial sums. The limiting sublinear law is represented as an upper expectation over stochastic integrals with a fractional Brownian moving-average kernel and predictable volatility controls, driven by a two-sided classical Brownian motion. When variance uncertainty vanishes, the limit reduces to the classical fractional Brownian limit. The proof combines a multidimensional Rosenthal inequality, coefficient approximation, and the invariance principle for independent and identically distributed random variables to establish finite-dimensional convergence. Tightness follows from second-moment increment bounds. Uniform approximation on compact sets then extends the convergence from bounded Lipschitz to bounded continuous test functions.
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BibTeX
@misc{gu2026invariance,
title = {{Invariance principle for long-memory linear processes under sublinear expectations}},
author = {Wang-Yun Gu and Li-Xin Zhang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02610},
eprint = {2611.02610},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02610},
}
RIS
TY - UNPB
TI - Invariance principle for long-memory linear processes under sublinear expectations
AU - Wang-Yun Gu
AU - Li-Xin Zhang
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02610
UR - https://arxiv.org/abs/2611.02610
AB - We establish an invariance principle for long-memory linear processes with power-law coefficients under sublinear expectations. Under truncated-moment and capacity-tail conditions, we construct these processes on a canonical space and prove weak convergence in continuous path space of their linearly interpolated, normalized partial sums. The limiting sublinear law is represented as an upper expectation over stochastic integrals with a fractional Brownian moving-average kernel and predictable volatility controls, driven by a two-sided classical Brownian motion. When variance uncertainty vanishes, the limit reduces to the classical fractional Brownian limit. The proof combines a multidimensional Rosenthal inequality, coefficient approximation, and the invariance principle for independent and identically distributed random variables to establish finite-dimensional convergence. Tightness follows from second-moment increment bounds. Uniform approximation on compact sets then extends the convergence from bounded Lipschitz to bounded continuous test functions.
ER -