A limit theorem for Bernoulli convolutions and the Φ-variation of functions in the Takagi class

2021/02/04 by Xiyue Han, Han, Xiyue, Alexander Schied +3
Mathematics · Economics, Econometrics and Finance · #Mathematical Dynamics and Fractals #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2102.02745

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 p^th 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.

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