This question comes from the following paper 1961(Blumenthal)

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 the Blumenthal-Getoor index is zero, we can not get 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 get a sufficient and necessary condition for Blumenthal-Gettor index = 0?