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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, 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?

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  • $\begingroup$ Condition in terms of... what? The definition already gives you a necessary and sufficient condition in terms of the Lévy measure, and conditions in terms of the characteristic exponent are well-known, too, so it is difficult to guess what kind of conditions you are looking for. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Dec 18, 2021 at 22:01
  • $\begingroup$ Let me make it more clear: I would like to find a specific class of Levy process that satisfies the sufficient and necessary condition in terms of $\beta =0$. $\endgroup$
    – Fractional analysics
    Commented Dec 18, 2021 at 22:16

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