This question comes from the following paper [1961(Blumenthal)][1]


  [1]: https://www.jstor.org/stable/24900735?seq=1#metadata_info_tab_contents
Let us consider a Levy process $X$ whose Levy triplet is $(a,s,\nu)$. According the above paper, Blumenthal-Gettor index is given by $$\beta=\inf\{\alpha>0:\int_{|x|<1}|x|^\alpha \nu(dx)<\infty\}.$$
 When the Levy process is finitely active, then we can get the Blumenthal-Getoor index is zero e.g. compound poisson process. While conversely, from the Blumenthal-Getoor index is zero, we can not get that Levy process is finitely active e.g. Gamma process(the Blumenthal-Getoor index of Gamma process is zero but it is infinitely active). Is it possible to find a specific class of Levy process that satisfies the sufficient and necessary condition in terms of Blumenthal-Gettor index $\beta$ = 0?