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

Abstract

We consider a probabilistic approach to compute the Wiener–Young Φ-variation of fractal functions in the Takagi class. Here, the 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 Φ we consider form a very wide class of functions that are regularly varying at zero. Moreover, for each such function Φ, our results provide in a straightforward manner a large and tractable class of functions that have nontrivial and linear Φ-variation. As a corollary, we also construct stochastic processes whose sample paths have nontrivial, deterministic, and linear Φ-variation for each function 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.