The classes of the $s$-selfdecomposable and decomposable probability measures are related to the limit distributions of sequences of random variables deformed by some nonlinear or linear transformations respectively. Both are characterized in many different ways, among others as distributions of some random integrals. In particular we get that each selfdecomposable probability measure is $s$-selfdecomposable. This and other relations between these two classes seem to be rather unexpected.
Zbigniew J. Jurek. "Relations Between the $s$-Selfdecomposable and Selfdecomposable Measures." Ann. Probab. 13 (2) 592 - 608, May, 1985. https://doi.org/10.1214/aop/1176993012