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

Xiyue Han,
Alexander Schied,
Zhenyuan Zhang

January, 2021

### Abstract

We consider Weierstraß and Takagi–van der Waerden functions with critical degree of roughness. In this case, the functions have vanishing th 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 $\mathit{\Phi}(x) = x / \sqrt{-\log x}$. 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.