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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 $(Z(x))_{x\in \Bbb{R}^{n}}$ with correlation function $c(h)=(1+\left\vert h\right\vert ^{\alpha })^{-\beta /\alpha },$ $h\in \Bbb{R}^{n}$, 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.

60G60Random fields
60G18Self-similar processes
62M40Statistics of random fields; image analysis
random fields
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