Packing dimension of the range of a Lévy process

Let $\{X(t)\}_{t\ge 0}$ denote a Lévy process in ${\mathbf{R}}^d$ with exponent $\Psi$. Taylor (1986) proved that the packing dimension of the range $X([0\,,1])$ is given by the index

$${(0.1)}\qquad\qquad \gamma' = \sup\left\{\alpha\ge 0: \liminf_{r ... ... \left\{\vert X(t)\vert \le r\right\}}{r^\alpha} \, dt =0\right\}.\qquad\qquad$$

We provide an alternative formulation of $\gamma'$ in terms of the Lévy exponent $\Psi$. Our formulation, as well as methods, are Fourier-analytic, and rely on the properties of the Cauchy transform. We show, through examples, some applications of our formula.