zbMATH — the first resource for mathematics

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
Full Text:
References:
 [1] Hawkes, J., A lower lipchitz condition for the stable subordinator, Z. wahr. verw. geb., 17, 23-32, (1971) · Zbl 0193.45002 [2] Hu, X.; Taylor, S.J., The multifractal structure of stable occupation measure, Stochastic process. appl., 66, 283-299, (1997) · Zbl 0888.28004 [3] Orey, S.; Taylor, S.J., How often on a Brownian path does the law of iterated logarithm fail?, Proc. London math. soc., 3, 174-192, (1974) · Zbl 0292.60128 [4] Perkins, E.A.; Taylor, S.J., Uniform measure results for the image of subsets under Brownian motion, Probab. th. rel. fields, 76, 257-289, (1987) · Zbl 0613.60071 [5] Taylor, S.J., Sample path properties of a transient stable processes, J. math. mech., 16, 1229-1246, (1967) · Zbl 0178.19301 [6] Taylor, S.J., The measure theory of random fractals, Math. proc. camb. phi. soc., 100, 383-406, (1986) · Zbl 0622.60021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.