On the connections between weakly stable and pseudo-isotropic distributions

B.H. Jasiulis and J.K. Misiewicz

Statistics & Probability Letters, 2008, vol. 78, issue 16, 2751-2755

Abstract: A random vector is weakly stable iff for all there exists a random variable [Theta] such that . 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 [Theta] such that where 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 [alpha]-norm for some [alpha]>0.

Date: 2008
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