I was not familiar with that paper but found it and took a quick look. About the same time it was published the Stanley group at BU published this more extensive analysis: https://www.researchgate.net/publication/11396961_Scaling_of_the_Distribution_of_Price_Fluctuations_of_Individual_Companies
It found the tail exponent of returns to be well outside the tail exponent range of Levy stable distributions.
The FractalDC algorithm will basically compress price data, preserving the extremes. So if you use it to preprocess price data, it should preserve the tail exponent, but at the same time it will remove data from the central portion of the return data so it creates a new distribution. If you are mainly interested in the tail exponent, you can just sort the data and select the tails, and get the same result. Data that arises from fractal processes do tend to have a power law tail behavior, but for stock markets in developed countries the tail exponent, α, is about 3.0.
It found the tail exponent of returns to be well outside the tail exponent range of Levy stable distributions.
The FractalDC algorithm will basically compress price data, preserving the extremes. So if you use it to preprocess price data, it should preserve the tail exponent, but at the same time it will remove data from the central portion of the return data so it creates a new distribution. If you are mainly interested in the tail exponent, you can just sort the data and select the tails, and get the same result. Data that arises from fractal processes do tend to have a power law tail behavior, but for stock markets in developed countries the tail exponent, α, is about 3.0.