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On the relationship between fractal dimension and fractal index for stationary stochastic processes. (English) Zbl 0798.60035
The real number $$d$$ is said to be the fractal dimension of a sample path on an interval of some stationary square-integrable stochastic process $$X_ t$$ if $$d$$ is the Hausdorff dimension of the path as a subset in $$\mathbb{R}^ 2$$. On the other hand the fractional index $$\alpha$$ of this process is defined by $\alpha = \sup \bigl\{ \beta \mid 1-\text{cov} (X_ 0, X_ t) = O (t^ \beta) \quad \text{as} \quad t \downarrow 0 \bigr\}.$ The authors give sufficient conditions that $$d=2 - \alpha/2$$ is valid provided that $$X_ t = g(Z_ t)$$ for a smooth function $$g$$ and $$Z_ t$$ is a stationary Gaussian process. It may be happen that $$X_ t$$ is non-Gaussian.

##### MSC:
 60G10 Stationary stochastic processes 60G15 Gaussian processes 62G05 Nonparametric estimation
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