Abstract
It is shown phenomenologically that the fractional derivative ξ = D α u of order α of a multifractal function has a power-law tail ∝\(\left| \xi \right|^{ - p_ \star}\) in its cumulative probability, for a suitable range of α's. The exponent is determined by the condition \(\zeta _{p_ \star } = {\alpha }p_ \star\), where ζ p is the exponent of the structure function of order p. A detailed study is made for the case of random multiplicative processes (Benzi et al., Physica D 65:352 (1993)) which are amenable to both theory and numerical simulations. Large deviations theory provides a concrete criterion, which involves the departure from straightness of the ζ p graph, for the presence of power-law tails when there is only a limited range over which the data possess scaling properties (e.g., because of the presence of a viscous cutoff). The method is also applied to wind tunnel data and financial data.
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Frisch, U., Matsumoto, T. On Multifractality and Fractional Derivatives. Journal of Statistical Physics 108, 1181–1202 (2002). https://doi.org/10.1023/A:1019843616965
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DOI: https://doi.org/10.1023/A:1019843616965