Variation and Hausdorff dimension of fractional Wiener-Weierstrass bridges
Abstract
Fractional Wiener--Weierstrass bridges are a class of Gaussian processes obtained by replacing trigonometric functions in the construction of classical Weierstrass functions by fractional Brownian bridges. A number of their sample path properties were derived in Schied--Zhang (2024,2026). The analysis in these papers left several open questions, most of which are addressed here. Specifically, we prove that, in the regime in which the Weierstrass mechanism dominates the underlying fractional Brownian bridge, the limiting b-adic variation coefficient has an absolutely continuous distribution and is therefore genuinely random. At the critical point between the two roughness regimes, we establish the power-variation formula and the critical Φ-variation limit conjectured in Schied--Zhang (2024). Finally, we derive the Hausdorff dimension for the graphs of the sample paths by proving a conjecture from Schied--Zhang (2026) for the missing high-Hurst case.
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BibTeX
@misc{schied2026variation,
title = {{Variation and Hausdorff dimension of fractional Wiener-Weierstrass bridges}},
author = {Alexander Schied and Zhenyuan Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.11994},
eprint = {2606.11994},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.11994},
}
RIS
TY - UNPB
TI - Variation and Hausdorff dimension of fractional Wiener-Weierstrass bridges
AU - Alexander Schied
AU - Zhenyuan Zhang
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2606.11994
UR - https://arxiv.org/abs/2606.11994
AB - Fractional Wiener--Weierstrass bridges are a class of Gaussian processes obtained by replacing trigonometric functions in the construction of classical Weierstrass functions by fractional Brownian bridges. A number of their sample path properties were derived in Schied--Zhang (2024,2026). The analysis in these papers left several open questions, most of which are addressed here. Specifically, we prove that, in the regime in which the Weierstrass mechanism dominates the underlying fractional Brownian bridge, the limiting b-adic variation coefficient has an absolutely continuous distribution and is therefore genuinely random. At the critical point between the two roughness regimes, we establish the power-variation formula and the critical Φ-variation limit conjectured in Schied--Zhang (2024). Finally, we derive the Hausdorff dimension for the graphs of the sample paths by proving a conjecture from Schied--Zhang (2026) for the missing high-Hurst case.
ER -