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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the $p$th variation of a class of fractal functions
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by Alexander Schied and Zhenyuan Zhang PDF
Proc. Amer. Math. Soc. 148 (2020), 5399-5412 Request permission

Abstract:

The concept of the $p$th variation of a continuous function $f$ along a refining sequence of partitions is the key to a pathwise It么 integration theory with integrator $f$. Here, we analyze the $p$th variation of a class of fractal functions, containing both the Takagi鈥搗an der Waerden and Weierstra脽 functions. We use a probabilistic argument to show that these functions have linear $p$th variation for a parameter $p\ge 1$, which can be interpreted as the reciprocal Hurst parameter of the function. It is shown, moreover, that if functions are constructed from (a skewed version of) the tent map, then the slope of the $p$th variation can be computed from the $p$th moment of a (non-symmetric) infinite Bernoulli convolution. Finally, we provide a recursive formula of these moments and use it to discuss the existence and non-existence of a signed version of the $p$th variation, which occurs in pathwise It么 calculus when $p\ge 3$ is an odd integer.
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Additional Information
  • Alexander Schied
  • Affiliation: Department of Statistics and Actuarial Science, University of Waterloo, Canada
  • Email: aschied@uwaterloo.ca
  • Zhenyuan Zhang
  • Affiliation: Department of Statistics and Actuarial Science, University of Waterloo, Canada
  • Email: z569zhan@uwaterloo.ca
  • Received by editor(s): July 2, 2019
  • Received by editor(s) in revised form: April 26, 2020
  • Published electronically: September 11, 2020
  • Additional Notes: The authors gratefully acknowledge financial support from the Natural Sciences and Engineering Research Council of Canada through grant RGPIN-2017-04054
  • Communicated by: Zhen-Qing Chen
  • © Copyright 2020 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 148 (2020), 5399-5412
  • MSC (2010): Primary 60H05, 28A80, 26A45, 60E05
  • DOI: https://doi.org/10.1090/proc/15171
  • MathSciNet review: 4163851