In this paper, it is shown that a random variable $Z$ having an infinitely divisible distribution with a finite left extremity can be decomposed in distribution as the sum of two random variables such that, given one, the other has an infinitely divisible distribution of prescribed type. Some extensions of this result when the left extremity is infinite are also given, and a new decomposition of a normal variable is exhibited.
"A New Decomposition of Infinitely Divisible Distributions." Ann. Probab. 3 (2) 338 - 345, April, 1975. https://doi.org/10.1214/aop/1176996404