Multifractality

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In mathematical analysis, multifractal analysis is the process of determining the fractal dimension of a multifractal system."
Multifractal analysis - Wikipedia

"A good price-fluctuations model should connect the behavior on multiple timescales. A natural test is the behavior of moments, in this case |rt|q as a function of q and t. Several groups report approximate power-law scaling with t, with different slopes for each value of q, as Figure 2 shows.21,22 In the jargon of dynamical systems theory, this suggests a fractal random process. A slope that is a linear function of q implies a simple fractal process, and a slope that is a nonlinear function of q implies a multifractal or multiscaling process. Indeed, several different calculations seem to show that the slope varies nonlinearly with q, suggesting that the price process is multifractal."
Farmer (1999)

"Thus, there is apparent multifractal behavior, even though asymptotically the process is just a simple fractal, with all the moments determined by the scaling of a single moment. For a short data set, a simple fractal process might look like a multifractal process due to slow convergence."
Farmer (1999)

"Our results suggest that it might be hard to distinguish apparent and true multifractal behavior in financial data."
Bouchaud, Potters and Meyer (2000)

"We find evidence of multifractality in the moment-scaling behavior of Deutsche Mark/US Dollar exchange rates."
Calvet and Fisher (2001)