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I have read that the Lévy–Khintchine representation exists for any infinitely divisible distribution. However, all the references I could find on Lévy–Khintchine representations are for Lévy processes. But how to derive the Lévy–Khintchine representation for a distribution, such as the Gamma distribution (not process)?

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    $\begingroup$ There is a one-to-one correspondence between infinitely divisible distributions and Levy processes. The Gamma distribution, for example, is the marginal of $X_{1}$, if $X$ is the Gamma process. See Proposition 2.1 here: galton.uchicago.edu/~lalley/Courses/385/LevyProcesses.pdf At any rate, the Levy-Kintchine representation for distributions is proved in Varadhan's book titled Probability Theory, if you have access to a library. $\endgroup$ – fourierwho Jan 2 '18 at 21:31

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