A Limit Theorem for Bernoulli Convolutions and the $$\Phi $$ Φ -Variation of Functions in the Takagi Class

We consider a probabilistic approach to compute the Wiener–Young $$\Phi $$ Φ -variation of fractal functions in the Takagi class. Here, the $$\Phi $$ Φ -variation is understood as a generalization of the quadratic variation or, more generally, the pth variation of a trajectory computed along the sequence of dyadic partitions of the unit interval. The functions $$\Phi $$ Φ we consider form a very wide class of functions that are regularly varying at zero. Moreover, for each such function $$\Phi $$ Φ , our results provide in a straightforward manner a large and tractable class of functions that have nontrivial and linear $$\Phi $$ Φ -variation. As a corollary, we also construct stochastic processes whose sample paths have nontrivial, deterministic, and linear $$\Phi $$ Φ -variation for each function $$\Phi $$ Φ 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.