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Sample Path Properties of the Fractional Wiener--Weierstrass Bridge

Alexander Schied, Zhenyuan Zhang

Abstract

Fractional Wiener--Weierstrass bridges are a class of Gaussian processes that arise from replacing the trigonometric function in the construction of classical Weierstrass functions by a fractional Brownian bridge. We investigate the sample path properties of such processes, including local and uniform moduli of continuity, $Φ$-variation, Hausdorff dimension, and location of the maximum. Our analysis relies heavily on upper and lower bounds of fractional integrals, where we establish a novel improvement of the classical Hardy--Littlewood inequality for fractional integrals of a special class of step functions.

Sample Path Properties of the Fractional Wiener--Weierstrass Bridge

Abstract

Fractional Wiener--Weierstrass bridges are a class of Gaussian processes that arise from replacing the trigonometric function in the construction of classical Weierstrass functions by a fractional Brownian bridge. We investigate the sample path properties of such processes, including local and uniform moduli of continuity, -variation, Hausdorff dimension, and location of the maximum. Our analysis relies heavily on upper and lower bounds of fractional integrals, where we establish a novel improvement of the classical Hardy--Littlewood inequality for fractional integrals of a special class of step functions.

Paper Structure

This paper contains 12 sections, 20 theorems, 196 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Notations.
  4. Outlook and some open questions.
  5. Some preliminary estimates
  6. Background on fractional integration
  7. A Hardy--Littlewood-type inequality for a sum of homogeneous indicator functions
  8. Covariance estimates
  9. Proofs of the main results
  10. -variation
  11. Moduli of continuity and Hausdorff dimension
  12. Distribution of the maximum location

Key Result

Theorem 2.2

Let $Y$ be a fractional Wiener--Weierstrass bridge with parameters $\alpha$, $b$, and $H$, and $K=\min\{1,({-\log_b\alpha})\}$. Moreover, in all three cases, if $\Theta:[0,\infty)\to[0,\infty)$ is a function such that $\Phi(x)=o(\Theta(x))$ as $x\to 0^+$, then $v_{\Theta}( Y )=\infty$ almost surely. Conversely, if $\Theta(x)=o(\Phi(x))$ as $x\to 0^+$, then $v_\Theta(Y)=0$ almost surely.

Theorems & Definitions (40)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Lemma 3.1: Corollary 1.9.2 of Mishura
  • Definition 3.2
  • Theorem 3.3
  • ...and 30 more