# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 ${\left(Z\left(x\right)\right)}_{x\in {ℝ}^{n}}$ with correlation function $c\left(h\right)={\left(1+{\left|h\right|}^{\alpha }\right)}^{-\beta /\alpha },$ $h\in {ℝ}^{n}$, where $\alpha \in \left(0,2\right]$ 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 processes 62M40 Statistics of random fields; image analysis 28A80 Fractals
random fields