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It is known since the 1970's that the Student's $t$-distribution is infinitely divisible. We can therefore apply the Lévy-Khintchine representation to it, and define the Lévy measure associated to a Student $t$-distribution.

Question: What is known about the Lévy measure of a Student's $t$-distribution? Do we have some closed form formula in terms of special functions? Is the behavior around $0$ or at $\infty$* known?

*Here is what I mean by this. If $\nu(\mathrm{d}t)$ is the Lévy measure, in the case of the Student's law, it has a density, meaning that $\nu(\mathrm{d}t) = \nu(t) \mathrm{d} t$. Now, what is the asymptotic behavior of à $\nu$ around $0$ or at $\infty$?

Any help would be appreciated.

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