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Multivariate multifractal analysis of L{é}vy functions. Part II: Validity of the multifractal formalism

Stéphane Jaffard, Lingmin Liao, Qian Zhang

TL;DR

The paper investigates the validity of the multivariate multifractal formalism for shifted Lévy functions by deriving the multivariate Legendre spectra using an oscillation-based approach. It rigorously distinguishes cases where the shift $y$ is a $b$-adic rational versus irrational, showing that the Legendre spectrum is confined to the diagonal for rational shifts and exhibits a dichotomy (sum versus min forms) depending on the boundedness of the digit-block sequence $\{w_k\}$ for irrational shifts. The results extend to $n$ functions and reveal that, for almost every $y$, the Legendre spectrum does not bound the true multivariate spectrum, highlighting fundamental limitations of the formalism in the multivariate setting. Numerical comparisons with wavelet-leader methods broadly align with the oscillation-based predictions, underscoring both the utility and the limitations of the Legendre formalism for practical multivariate multifractal analysis of non-stationary signals.

Abstract

In this article, we determine the multivariate multifractal Legendre spectra of shifted L{é}vy functions. This allows us to explore how the validity of the multivariate multifractal formalism depends on the shift parameter. This article is a continuation of [10] where the corresponding bivariate multifractal spectra were explored.

Multivariate multifractal analysis of L{é}vy functions. Part II: Validity of the multifractal formalism

TL;DR

The paper investigates the validity of the multivariate multifractal formalism for shifted Lévy functions by deriving the multivariate Legendre spectra using an oscillation-based approach. It rigorously distinguishes cases where the shift is a -adic rational versus irrational, showing that the Legendre spectrum is confined to the diagonal for rational shifts and exhibits a dichotomy (sum versus min forms) depending on the boundedness of the digit-block sequence for irrational shifts. The results extend to functions and reveal that, for almost every , the Legendre spectrum does not bound the true multivariate spectrum, highlighting fundamental limitations of the formalism in the multivariate setting. Numerical comparisons with wavelet-leader methods broadly align with the oscillation-based predictions, underscoring both the utility and the limitations of the Legendre formalism for practical multivariate multifractal analysis of non-stationary signals.

Abstract

In this article, we determine the multivariate multifractal Legendre spectra of shifted L{é}vy functions. This allows us to explore how the validity of the multivariate multifractal formalism depends on the shift parameter. This article is a continuation of [10] where the corresponding bivariate multifractal spectra were explored.
Paper Structure (7 sections, 10 theorems, 34 equations, 4 figures)

This paper contains 7 sections, 10 theorems, 34 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Statement of the main results
  3. Multivariate multifractal formalism
  4. Proof of Theorem \ref{['multitheofinal']}
  5. Proof of Theorem \ref{['multivariatemultifractalLegendrespectrum_nfunctionsrational']}
  6. Proof of Theorem \ref{['multivariatemultifractalLegendrespectrum_nfunctions']}
  7. Concluding remarks and numerical illustrations

Key Result

Theorem 2.1

jaffardliaoqian Let $\alpha_1, \alpha_2>0$. For almost every $y$, the bivariate multifractal spectrum $\mathcal{D}_{L_{\alpha_1}^b,L_{\alpha_2}^{b,y}}$ of the Lévy functions $L_{\alpha_1}^{b}$ and $L_{\alpha_2}^{b,y}$, is given by

Figures (4)

  • Figure 5: Scaling function ($y$ is $b$-adic rational)
  • Figure 6: Scaling function ($y$ is non-$b$-adic rational)
  • Figure 8: Numerical results of the bivariate multifractal Legendre spectrum $\mathcal{L}_{L_{0.3}^2,L_{0.7}^{2,y}}$ for $y = 1/3, 1/5, 1/2 , 1/4$ based on oscillations vs. wavelet leaders ). The left four bivariate multifractal Legendre spectra are based on order 2 oscillations and the right four are based on wavelet leaders. We also show the projections of the computed bivariate spectra on the planes $H_1 = 0$ and $H_2 =0$ which, theoretically, yield the corresponding univariate spectra (see jaffard2019multifractal)
  • Figure 9: The left two: the difference between the oscillation Legendre spectrum and the theoretical result based on the oscillations of bivariate Legendre spectra \ref{['1/3']}, with $y=1/3$ and $1/5$ respectively. The right two: the difference between the wavelet leader Legendre spectrum and the theoretical result with $y=1/3$ and $1/5$ respectively.

Theorems & Definitions (14)

  • Definition 1.1
  • Definition 1.2
  • Theorem 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Remark 1
  • Remark 2
  • Theorem 2.4
  • Theorem 2.5
  • Proposition 3.1
  • ...and 4 more