Abstract
In this chapter we return to mathematical foundations and discuss the theory of stable distributions as an extension to the central limit theorem and the canonical representation of stable distributions. The solution of the one-dimensional Weierstrass random walk is presented in detail, including its fractal properties and super-diffusive behavior. Fractal time random walks and their relation to subdiffusive behavior are introduced next. Finally, we discuss the truncated Lévy flight which will be employed in Chap. 5 for the description of financial fluctuations.
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Notes
- 1.
This is not the only definition of a fractal dimension. There are other methods. However, provided certain mathematical premises, one can establish that all definitions agree with one another [66]. Here, we take this for granted and only mention the Hausdorff dimension, which is the most intuitive one.
- 2.
Note that the terms ‘random flight’ and ‘random walk’ are sometimes used to distinguish a detail of the stochastic process. One speaks of a random flight if the process occurs in continuous space and consists of a sequence of independent displacements with random direction and magnitude (but finite second moment). On the other hand, if the displacements are restricted to a lattice, a random walk results. Usually, we do not make this distinction and laxly use random flights and random walks synonymously.
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Paul, W., Baschnagel, J. (2013). Beyond the Central Limit Theorem: Lévy Distributions. In: Stochastic Processes. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00327-6_4
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