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Typical Martingale Diverges at a Typical Point

Author

Listed:
  • Ondřej F. K. Kalenda

    (Charles University)

  • Jiří Spurný

    (Charles University)

Abstract

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

Suggested Citation

  • Ondřej F. K. Kalenda & Jiří Spurný, 2016. "Typical Martingale Diverges at a Typical Point," Journal of Theoretical Probability, Springer, vol. 29(1), pages 180-205, March.
    • Handle: RePEc:spr:jotpro:v:29:y:2016:i:1:d:10.1007_s10959-014-0567-7
    DOI: 10.1007/s10959-014-0567-7
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