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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
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