Journal of Theoretical Probability

, Volume 29, Issue 1, pp 180–205 | Cite as

Typical Martingale Diverges at a Typical Point

  • Ondřej F. K. Kalenda
  • Jiří Spurný


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.


\(L^1\)-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.


  1. 1.
    Belna, C.L., Evans, M.J., Humke, P.D.: Symmetric and ordinary differentiation. Proc. Am. Math. Soc. 72(2), 261–267 (1978)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Calude, C., Zamfirescu, T.: Most numbers obey no probability laws. Publ. Math. Debrecen 54(suppl), 619–623 (1999). Automata and formal languages, VIII (Salgótarján. 1996)Google Scholar
  3. 3.
    Dolženko, E.P.: Boundary properties of arbitrary functions. Izv. Akad. Nauk SSSR Ser. Mat. 31, 3–14 (1967)MathSciNetGoogle Scholar
  4. 4.
    Dougherty, R., Mycielski, J.: The prevalence of permutations with infinite cycles. Fundam. Math. 144(1), 89–94 (1994)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Fremlin, D.H.: Measure Theory. Vol. 2. Torres Fremlin, Colchester, (2003). Broad foundations, Corrected second printing of the 2001 originalGoogle Scholar
  6. 6.
    Fremlin, D.H.: Measure Theory. Vol. 3. Torres Fremlin, Colchester, (2004). Measure algebras, Corrected second printing of the 2002 originalGoogle Scholar
  7. 7.
    Kechris, A.S.: Classical Descriptive Set Theory, vol. 156 of Graduate Texts in Mathematics. Springer, New York (1995)Google Scholar
  8. 8.
    Lindenstrauss, J., Preiss, D., Tišer, J.: Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces, vol. 179 of Annals of Mathematics Studies. Princeton University Press, Princeton (2012)Google Scholar
  9. 9.
    Long, R.L.: Martingale Spaces and Inequalities. Peking University Press, Beijing (1993)CrossRefzbMATHGoogle Scholar
  10. 10.
    Matoušek, J., Matoušková, E.: A highly non-smooth norm on Hilbert space. Isr. J. Math. 112, 1–27 (1999)CrossRefzbMATHGoogle Scholar
  11. 11.
    Matoušková, E.: An almost nowhere Fréchet smooth norm on superreflexive spaces. Stud. Math. 133(1), 93–99 (1999)zbMATHGoogle Scholar
  12. 12.
    Oxtoby, J.C.: Measure and Category, second edn., vol. 2 of Graduate Texts in Mathematics. Springer, New York (1980). A survey of the analogies between topological and measure spacesGoogle Scholar
  13. 13.
    Preiss, D., Tišer, J.: Two unexpected examples concerning differentiability of Lipschitz functions on Banach spaces. In: Geometric Aspects of Functional Analysis (Israel, 1992–1994), vol. 77 of Oper. Theory Adv. Appl. Birkhäuser, Basel, pp. 219–238 (1995)Google Scholar
  14. 14.
    Spurný, J., Zelený, M.: Convergence of a typical martingale (a remark on the Doob theorem). J. Math. Anal. Appl. 414(2), 945–958 (2014)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Troitsky, V.G.: Martingales in Banach lattices. Positivity 9(3), 437–456 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Zajíček, L.: On differentiability properties of typical continuous functions and Haar null sets. Proc. Am. Math. Soc. 134(4), 1143–1151 (2006). (electronic)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Mathematical Analysis, Faculty of Mathematics and PhysicsCharles UniversityPrague 8Czech Republic

Personalised recommendations