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

Abstract

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 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 $\Phi$ we consider form a very wide class of functions that are regularly varying at zero. Moreover, for each such function $\Phi$, our results provide in a straightforward manner a large and tractable class of functions that have nontrivial and linear $\Phi$-variation. As a corollary, we also construct stochastic processes whose sample paths have nontrivial, deterministic, and linear $\Phi$-variation for each function $\Phi$ 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.