Packing and covering indices for a general Lévy process

Abstract

There has been substantial interest in the indices $0 \leq \beta'' \leq \beta' \leq \beta \leq 2$, defined by Blumenthal and Getoor, determined by a general Lévy process in $\mathbf{R}^d$. Pruitt defined an index $\gamma$ which determines the covering dimension and Taylor showed that an index $\gamma'$, first considered by Hendricks, determines the packing dimension for the trajectory. In the present paper we prove that

$$\frac{\beta}{2} \le \gamma' \le \min(\beta, d),

and give examples to show that the whole range is attainable. However, we cannot completely determine the set of values of $(\gamma, \gamma', \beta)$ which can be attained as indices of some Lévy process.