A probabilistic approach to the $\Phi$-variation of classical fractal functions with critical roughness

TL;DR

This paper investigates the $ ext{Φ}$-variation of classical fractal functions at critical roughness, revealing finite, nonzero variation along specific partitions, despite their nowhere differentiability and vanishing $p$-variation.

Contribution

It demonstrates that Weierstraß and Takagi-van der Waerden functions have finite, nonzero $ ext{Φ}$-variation at critical roughness, using advanced probabilistic limit theorems.

Findings

$ ext{Φ}$-variation is finite and nonzero for these functions.

The variation is linear along $b$-adic partitions.

Different CLTs are applied depending on the function and parameter parity.

Abstract

We consider Weierstra\ss\ and Takagi-van der Waerden functions with critical degree of roughness. In this case, the functions have vanishing $p^{\text{th}}$ 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 $\Phi$-variation along the sequence of $b$-adic partitions, where $\Phi(x)=x/\sqrt{-\log x}$. For the Weierstra\ss\ 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.