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