LAUSR

Abstract In 1938, Schoenberg [20] posed a few seemingly simple problems in the area of elementary probability theory. The main goal was to characterize all the pseudo-isotropic distributions; that is, probability distributions in which all the one-dimensional projections are the same up to a scale parameter c. However, except for rotationally invariant and symmetric stable distributions this goal turned out to be extremely difficult and significant results appeared after four decades. In 2005, Misiewicz, Oleszkiewicz and Urbanik [17] presented the introduction to the theory of weakly-stable distributions and vectors, where a random vector X, taking values in a Banach space E , is called weakly-stable iff for all random variables Q 1 , Q 2 there exists a random variable Θ such that ( ⁎ ) X Q 1 + X ′ Q 2 = d X Θ , where X , X ′ , Q 1 , Q 2 , Θ are independent. In [6] , the authors showed that under some additional weak assumptions, every extreme point of the set of pseudo-isotropic distributions with the fixed quasi-norm c has to be weakly-stable. In this paper we show the converse implication; i.e. we show that every symmetric weakly-stable random vector is pseudo-isotropic. This seems to be another small step in solving Schoenberg's problems. As an application of our results, we propose a method to check whether a given symmetric multi-dimensional distribution is pseudo-isotropic, or, equivalently, whether or not a given symmetric weakly-stable distribution has a multidimensional version.