Typical martingale diverges at a typical point

Abstract

We investigate convergence of martingales adapted to a given filtration of finite \(\sigma\)-algebras. To any such filtration we associate a canonical metrizable compact space \(K\) such that martingales adapted to the filtration can be canonically represented on \(K\). We further show that (except for trivial cases) typical martingale diverges at a comeager subset of \(K\). `Typical martingale' means a martingale from a comeager set in any of the standard spaces of martingales. In particular we show that a typical \(L^1\)-bounded martingale of norm at most one converges almost surely to zero and has maximal possible oscillation on a comeager set.