Tweedie Distributions - The Tweedie Models and Multifractality

The Tweedie Models and Multifractality

From the properties of self-similar processes, the power-law exponent p=2-d is related to the Hurst exponent H and the fractal dimension D by

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A one-dimensional data sequence of self-similar data may demonstrate a variance-to-mean power law with local variations in the value of p and hence in the value of D. When fractal structures manifest local variations in fractal dimension, they are said to be multifractals. Examples of data sequences that exhibit local variations in p like this include the eigenvalue deviations of the Gaussian Orthogonal and Unitary Ensembles. The Tweedie compound Poisson–gamma distribution has served to model multifractality based on local variations in the Tweedie exponent α. Consequently, in conjunction with the variation of α, the Tweedie convergence theorem can be viewed as having a role in the genesis of such multifractals.