Fig. 4

In the regime \(\mathbb {S}_{2}=\{(p,q): 0< p=q<1 \}\) (the dashed line) ϕ_(q,p,α)(t) is self-decomposable. We also emphasize the point (p,q)=(2,2) (the black square) corresponds to the Gaussian characteristic function, and the point (p,q)=(1,1) (the black circle) corresponds to the Laplace characteristic function. The regime \(\mathbb {S}_{1}=\{ (q,p): 2< q < p \}\) (the gray triangle) is where ϕ_(q,p,α)(t) is not a characteristic function for almost all α≥1. The white space is the regime where ϕ_(q,p,α)(t) is not a characteristic function for all α≥1