On the rate of convergence for Takagi class functions

  • Shoto Osaka
  • Masato TakeiEmail author
Original Paper


We consider a generalized version of the Takagi function, which is one of the most famous example of nowhere differentiable continuous functions. We investigate a set of conditions to describe the rate of convergence of Takagi class functions from the probabilistic point of view: The law of large numbers, the central limit theorem, and the law of the iterated logarithm. On the other hand, we show that the Takagi function itself does not satisfy the law of large numbers in the usual sense.


Nowhere differentiable continuous functions Probabilistic method Limit theorems Martingales 

Mathematics Subject Classification

60F05 26A27 60G42 60G46 



The authors deeply thank anonymous referees for their valuable comments. In particular, one of referees suggests simplification of some of our original arguments, which are adopted in the revised version. They also thank the editor for helpful comments.


  1. 1.
    Allaart, P.C.: On a flexible class of continuous functions with uniform local structure. J. Math. Soc. Jpn. 61, 237–262 (2009)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Allaart, P.C., Kawamura, K.: The Takagi function: a survey. Real Anal. Exch. 37, 1–54 (2011/2012)Google Scholar
  3. 3.
    Azuma, K.: Weighted sums of certain dependent random variables. Tôhoku Math. J. 2(19), 357–367 (1967)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Billingsley, P.: Ergodic Theory and Information. Wiley, New York (1965)zbMATHGoogle Scholar
  5. 5.
    Gamkrelidze, N.G.: On a probabilistic properties of Takagi’s function. J. Math. Kyoto Univ. 30, 227–229 (1990)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hall, P., Heyde, C.C.: Martingale Limit Theory and its Application, Probability and Mathematical Statistics. Academic Press, New York (1980)zbMATHGoogle Scholar
  7. 7.
    Hata, M., Yamaguti, M.: The Takagi function and its generalization. Jpn. J. Appl. Math. 1, 183–199 (1984)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Heyde, C.C.: On central limit and iterated logarithm supplements to the martingale convergence theorem. J. Appl. Probab. 14, 758–775 (1977)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kôno, N.: On generalized Takagi functions. Acta Math. Hung. 49, 315–324 (1987)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Lagarias, J.C.: The Takagi function and its properties, functions in number theory and their probabilistic aspects. RIMS Kôkyûroku Bessatsu B34, 153–189 (2012)zbMATHGoogle Scholar
  11. 11.
    Scott, D.J., Huggins, R.M.: On the embedding of processes in Brownian motion and the law of the iterated logarithm for reverse martingales. Bull. Aust. Math. Soc. 27, 443–459 (1983)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Stout, W.F.: Almost sure convergence, Probability and Mathematical Statistics, vol. 24. Academic Press, New York (1974)Google Scholar
  13. 13.
    Takagi, T.: A simple example of a continuous function without derivative. Proc. Phys. Math. Jpn. 1, 176–177 (1903)zbMATHGoogle Scholar
  14. 14.
    Yamaguti, M., Hata, M.: Weierstrass’s function and chaos. Hokkaido Math. J. 12, 333–342 (1983)MathSciNetCrossRefGoogle Scholar

Copyright information

© The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Graduate School of Engineering ScienceYokohama National UniversityYokohamaJapan
  2. 2.Department of Applied Mathematics, Faculty of EngineeringYokohama National UniversityYokohamaJapan

Personalised recommendations