A probabilistic approach to the Φ-variation of classical fractal functions with critical roughness

Xiyue Han, Alexander Schied and Zhenyuan Zhang

Statistics & Probability Letters, 2021, vol. 168, issue C

Abstract: We consider Weierstraß and Takagi–van der Waerden functions with critical degree of roughness. In this case, the functions have vanishing pth variation for all p>1 but are also nowhere differentiable and hence not of bounded variation either. We resolve this apparent puzzle by showing that these functions have finite, nonzero, and linear Wiener–Young Φ-variation along the sequence of b-adic partitions, where Φ(x)=x∕−logx. For the Weierstraß functions, our proof is based on the martingale central limit theorem (CLT). For the Takagi–van der Waerden functions, we use the CLT for Markov chains if a certain parameter b is odd, and the standard CLT for b even.

Keywords: Weierstraß function; Takagi–van der Waerden functions; Wiener–Young Φ-variation; Martingale central limit theorem; Markov chain central limit theorem (search for similar items in EconPapers)
Date: 2021
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Citations: View citations in EconPapers (2)

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DOI: 10.1016/j.spl.2020.108920

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