On the connections between weakly stable and pseudo-isotropic distributions

A random vector X is weakly stable iff for all a,b∈R there exists a random variable Θ such that aX+bX′=dXΘ. This is equivalent (see Misiewicz et al. [Misiewicz, J.K., Oleszkiewicz, K., Urbanik, K., 2005. Classes of measures closed under mixing and convolution. Weak stability. Stud. Math. 167 (3), 195-213]) to the condition that for all random variables Q1,Q2 there exists a random variable Θ such that (∗ )XQ1+X′Q2=dXΘ, where X,X′,Q1,Q2,Θ are independent. Some of the weakly stable distributions turn out to be the extreme points for the class of pseudo-isotropic distributions, where the distribution is pseudo-isotropic if all its one-dimensional projections are the same up to a scale parameter. We show here that the scaling function for pseudo-isotropic distribution can define a generalized distribution iff it is an α-norm for some α>0.