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

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Listed:
  • Xiyue Han

    (University of Waterloo)

  • Alexander Schied

    (University of Waterloo)

  • Zhenyuan Zhang

    (Stanford University)

Abstract

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.

Suggested Citation

  • Xiyue Han & Alexander Schied & Zhenyuan Zhang, 2022. "A Limit Theorem for Bernoulli Convolutions and the $$\Phi $$ Φ -Variation of Functions in the Takagi Class," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2853-2878, December.
  • Handle: RePEc:spr:jotpro:v:35:y:2022:i:4:d:10.1007_s10959-022-01157-1
    DOI: 10.1007/s10959-022-01157-1
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    References listed on IDEAS

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    1. Jim Gatheral & Thibault Jaisson & Mathieu Rosenbaum, 2018. "Volatility is rough," Quantitative Finance, Taylor & Francis Journals, vol. 18(6), pages 933-949, June.
    2. Han, Xiyue & Schied, Alexander & Zhang, Zhenyuan, 2021. "A probabilistic approach to the Φ-variation of classical fractal functions with critical roughness," Statistics & Probability Letters, Elsevier, vol. 168(C).
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