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We consider a probabilistic approach to compute the Wiener--Young$\Phi$-variation of fractal functions in the Takagi class. Here, the$\Phi$-variation is understood as a generalization of the quadratic variationor, more generally, the $p^{\text{th}}$ variation of a trajectory computedalong the sequence of dyadic partitions of the unit interval. The functions$\Phi$ we consider form a very wide class of functions that are regularlyvarying at zero. Moreover, for each such function $\Phi$, our results providein a straightforward manner a large and tractable class of functions that havenontrivial and linear $\Phi$-variation. As a corollary, we also constructstochastic processes whose sample paths have nontrivial, deterministic, andlinear $\Phi$-variation for each function $\Phi$ from our class. The proof ofour main result relies on a limit theorem for certain sums of Bernoulli randomvariables that converge to an infinite Bernoulli convolution.

A limit theorem for Bernoulli convolutions and the $\Phi$-variation of functions in the Takagi class