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

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