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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
Keywords:
random fields
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