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Consider an infinitely divisible random variable X defined on $\mathbb{R}$ with Levy triple $(a,\sigma,\nu)$ following from the LK representation.

Can all or any infinitely divisible random variables be decomposed into the sum of two independent random variables Y & Z such that X = Y + Z where Y is an inf div r.v defined on $(-\infty,0)$ and Z is an inf div r.v defined on $[0,\infty)$?

It seems that the converse holds automatically that if Y & Z are inf div then so is X = Z + Y.

Equivalently (I think) can any Levy process with positive and negative jumps be decomposed into the sum of two processes; one Levy process with only positive jumps and another Levy process with only negative jumps?

Thanks

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    the answer to the last is no. Increasing Levy processes are called 'subordinators', and they, of course, must be of bounded variation which only happens if \int_0^1 x \nu(dx) < \infty ,\;(\nu the Levy measure). For example, you cannot find a subordinator that scales like the Cauchy.2012-05-23

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