Typical Martingale Diverges on a Co-$\sigma$-porous Set

Abstract

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-$\sigma$-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-$\sigma$-porous set with respect to the first metric, no martingale diverges on a co-$\sigma$-porous set with respect to the second metric. Finally, we investigate the key factors determining whether the divergence set can be co-$\sigma$-porous.