Multifractal

Lesson

Introduction
Multifractal theory
Hierarchy of singularities
Multifractal and extremes
- Extremes and multifractal
- Divergence of statistical moment
- Sample's size effect
Universal multifractal model
Exercises
Conclusion

Divergence of statistical moment

The strong variability of the process at small scale compared to the one at large scale, induce to the divergence of order larger than the value :

$E(\varepsilon^{q}_{\lambda}) \rightarrow \infty$

Where is the divergence moment order. When we measure some process, we can expect that the observation are made at the resolution which can not be the the resolution in which the phenomenon expresses the whole variability. Indeed when we go through the finest scale we can observe a strong variability. Intuitively we can suppose the observation at the resolution smooths the variability of the process at the resolution .

According to the previous equation, the statistical moment function takes the form of for . Following the Legendre transform we can obtain:

$K(q) = \gamma_{max}(q-q_D) + K(q_D) \quad for \quad q >q_D \\ \quad \\ c(\gamma) = q_D (\gamma-\gamma_D) + c(\gamma_D) \quad for \quad \gamma > \gamma_D$

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