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For infinitely divisible random variables, Blumenthal and Getoor introduced in [1] an index that allow to study for instance the local Hölder regularity of Lévy processes.

For a symmetric infinitely divisible random variable $X$ with characteristic exponent $\psi$ and Lévy measure $\nu$, we have in particular \begin{align} \beta &= \inf \{ p>0; \frac{\psi(t)}{\lvert t \rvert^p} \underset{\lvert t \rvert \rightarrow \infty}{\longrightarrow} 0 \} \\ &= \inf \{ \alpha >0 ; \int_{|x|< 1} |x|^\alpha \nu (\mathrm{d}x) <\infty\}. \end{align} Knowing the Lévy measure, it is in general straightforward to find $\beta$. However, for some infinitely divisible laws, the Lévy measure does not seem to be known. Typically, for the best of my knowledge, this is the case for the student $t$-distribution $$p_v(x) \propto (1+x^2)^{-\frac{v+1}{2}}, $$ which is shown to be infinitely divisible (cf. [2]). Is the Blumenthal-Getoor index of $p_v$ known?

[1] R. M. Blumenthal and R. K. Getoor, Sample functions of stochastic processes with stationary independent increments, J. Math. Mech. 10 (1961), pp. 493–516

[2] M.E.H. Ismail, Bessel functions and the infinitely divisibility of the Student $t$-distribution, The Annals of Probability, Vol. 5, N. 4 (Aug. 1977), pp. 582-585

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  • $\begingroup$ The characteristic function of the Student's t-distribution is proportional to $K_{\nu/2}(\sqrt{\nu}|t|) |t|^{\nu/2}$. The logarithm of the above is the characteristic exponent $-\psi(t)$. The behaviour of $\psi(t)$ at infinity is therefore roughly comparable with $t$. Therefore, the BG index should be equal to $1$. (This is consistent with the fact that that Student t-distribution reduces to the Cauchy distribution if $\nu = 1$). $\endgroup$
    – Mateusz Kwaśnicki
    May 13, 2017 at 22:15

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