典型鞅在共σ-多孔集上发散
Typical Martingale Diverges on a Co-$σ$-porous Set
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中文总结 AI 辅助
该研究针对康托尔空间上的L^p有界鞅，证明在两种标准度量下典型鞅的发散性与共σ-多孔集的关联，探究决定发散集能否为共σ-多孔集的关键因素。
中文摘要 AI 辅助
我们考虑康托尔空间上p∈[1,∞]时的L^p有界鞅空间，为康托尔空间配备两种标准度量，证明在两种度量下，典型鞅都在典型点发散。此处“典型鞅”指p∈[1,∞)时属于共σ-多孔集的元素，p=∞时属于共多孔集的元素。具体而言，我们证明在第一种度量下，典型鞅在共σ-多孔集上发散；而在第二种度量下，没有鞅会在共σ-多孔集上发散。最后，我们探究决定发散集能否为共σ-多孔集的关键因素。
英文摘要
We consider the space of $L^p$-bounded martingales on the Cantor space for $p\in[1,\infty]$. 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 $p\in[1,\infty)$ and of a co-porous set when $p=\infty$. 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.
发表机构
- Charles University（查理大学）
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