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Zbl 1062.60053
Gneiting, Tilmann; Schlather, Martin
Stochastic models that separate fractal dimension and the Hurst effect.
(English)
[J] SIAM Rev. 46, No. 2, 269-282 (2004). ISSN 0036-1445; ISSN 1095-7200/e

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.
[Marius Iosifescu (Bucureşti)]
MSC 2000:
*60G60 Random fields
60G18 Self-similar processes
62M40 Statistics of random fields
28A80 Fractals

Keywords: random fields

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