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Logarithmic multifractal spectrum of stable occupation measure. (English) Zbl 0932.60041
Summary: For a stable subordinator $$Y_t$$ of index $$\alpha$$, $$0<\alpha < 1$$, the occupation measure $$\mu (A) = |\{t\in [0,1] : Y_t \in A\}|$$ is known to have (with probability 1) the property that $\liminf_{r\downarrow 0} \frac {\ln \mu (x-r,x+r)}{\ln r} = \alpha , \quad \forall x\in Y[0,1].$ In order to obtain an interesting spectrum for the large values of $$\mu (x-r,x+r)$$, we consider the set $B_{\theta} = \Big {\{} x\in Y[0,1]:\limsup_{r\downarrow 0} \frac {\mu (x-r,x+r)}{c_{\alpha} r^{\alpha} (\ln (1/r))^{1-\alpha}} = \theta \Big {\}},$ where $$c_{\alpha}$$ is a suitable constant. It is shown that $$B_{\theta} = \emptyset$$ for $$\theta >1$$, and $$B_{\theta} \not = \emptyset$$ for $$0\leq \theta \leq 1$$; moreover, dim $$B_{\theta} = \text{Dim } B_{\theta} = \alpha (1-\theta ^{1/(1-\alpha)})$$.

##### MSC:
 60G17 Sample path properties
##### Keywords:
stable subordinators; multifractals; occupation measures
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##### References:
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