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On the rate of convergence for Takagi class functions

  • Shoto Osaka
  • Masato TakeiEmail author
Original Paper
  • 9 Downloads

Abstract

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.

Keywords

Nowhere differentiable continuous functions Probabilistic method Limit theorems Martingales 

Mathematics Subject Classification

60F05 26A27 60G42 60G46 

Notes

Acknowledgements

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.

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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

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