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Typical Martingale Diverges on a Co-σ-porous Set
Abstract
We consider the space of Lp-bounded martingales on the Cantor space for p∈[1,∞]. Equipping the Cantor space with two standard metrics, we show that under both, a typical martingale diverges at a typical point. By a 'typical martingale', we mean an element of a co-σ-porous set when p∈[1,∞) and of a co-porous set when p=∞. Specifically, we show that while a typical martingale diverges on a co-σ-porous set with respect to the first metric, no martingale diverges on a co-σ-porous set with respect to the second metric. Finally, we investigate the key factors determining whether the divergence set can be co-σ-porous.
Related
BibTeX
@misc{hejny2026typical,
title = {{Typical Martingale Diverges on a Co-$σ$-porous Set}},
author = {Antonín Hejný},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03793},
eprint = {2609.03793},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03793},
}
RIS
TY - UNPB
TI - Typical Martingale Diverges on a Co-σ-porous Set
AU - Antonín Hejný
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03793
UR - https://arxiv.org/abs/2609.03793
AB - We consider the space of L^p-bounded martingales on the Cantor space for p∈[1,∞]. Equipping the Cantor space with two standard metrics, we show that under both, a typical martingale diverges at a typical point. By a 'typical martingale', we mean an element of a co-σ-porous set when p∈[1,∞) and of a co-porous set when p=∞. Specifically, we show that while a typical martingale diverges on a co-σ-porous set with respect to the first metric, no martingale diverges on a co-σ-porous set with respect to the second metric. Finally, we investigate the key factors determining whether the divergence set can be co-σ-porous.
ER -