Typical Martingale Diverges at a Typical Point
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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.
Keywords\(L^1\)-bounded martingale \(L^p\)-bounded martingale Filtration of finite \(\sigma \)-algebras Oscillation Comeager set
Mathematics Subject Classification (2010)60G42 54E52 54E70
We are grateful to the referees for their helpful comments which we used to improve the presentation of the paper. Our research was supported by Grant GAČR P201/12/0290. The second author was also supported by The Foundation of Karel Janeček for Science and Research.
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