There are two types of generalizations of selfdecomposability of probability measures on $\mathbf{R}^d, d\geq 1$ : the $c$-decomposability and the $C$-decomposability of Loève and Bunge on the one hand, and the semi-selfdecomposability of Maejima and Naito on the other.
The latter implies infinite divisibility but the former does not in general.
For $d\geq 2$ introduction of operator (matrix) normalizations yields four kinds of classes of distributions on $\mathbf{R}^d : L_{0}(b,Q),\tilde{L}_{0}(b,Q),L_{0}(C,Q)$, and $\tilde{L}_{0}(C,Q)$, where $0