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

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.