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Stochastic models that separate fractal dimension and the Hurst effect. (English) Zbl 1062.60053
The authors’ aim is to introduce stochastic models allowing arbitrary combinations of fractal dimension $$D$$ and Hurst coefficient $$H$$ which characterizes long-memory dependence. For self-affine models in $$n$$-dimensional space such as fractional Brownian motion one has $$D+H=n+$$1. The authors’ key item is the Cauchy class consisting of the stationary Gaussian random fields $$(Z(x))_{x\in \mathbb{R}^{n}}$$ with correlation function $$c(h)=(1+\left| h\right| ^{\alpha })^{-\beta /\alpha },$$ $$h\in \mathbb{R}^{n}$$, where $$\alpha \in (0,2]$$ and $$\beta >0$$. This simple model allows any combination of the two parameters $$D$$ and $$H.$$ Two figures provide displays of profiles and images in which the effects of fractal dimension and Hurst coefficient are decoupled. Special attention is paid to the problem of estimating $$D$$ and $$H$$ when the equation $$D+H=n+1$$ does not hold. Related models able to separate fractal dimension and Hurst effect are also discussed.

##### MSC:
 60G60 Random fields 60G18 Self-similar stochastic processes 62M40 Random fields; image analysis 28A80 Fractals
random fields
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