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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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