The multifractal structure of stable occupation measure

Let X be a stable subordinator of index [alpha] and [mu] be the occupation measure of X. Denote d([mu],x) and as the lower and upper local dimensions of [mu]. We obtain that the Hausdorff dimension of the set of the points where is (2[alpha]2/[beta]) - [alpha] a.s. and the lower bound of packing dimension is 2[alpha] - [beta] a.s. if [alpha][less-than-or-equals, slant][beta][less-than-or-equals, slant]2[alpha]. When [beta] > 2[alpha], the corresponding set is empty a.s.. And for a.s. [Omega], the set of the points where is the closure of X[0,1].