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.
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Notes
It was shown to us by Professor Fremlin that the set \(M_{1,1}\cap M_1^s\) is of type \(F_{\sigma \delta }\).
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Acknowledgments
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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Kalenda, O.F.K., Spurný, J. Typical Martingale Diverges at a Typical Point. J Theor Probab 29, 180–205 (2016). https://doi.org/10.1007/s10959-014-0567-7
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DOI: https://doi.org/10.1007/s10959-014-0567-7
Keywords
- \(L^1\)-bounded martingale
- \(L^p\)-bounded martingale
- Filtration of finite \(\sigma \)-algebras
- Oscillation
- Comeager set