Some remarks on S alpha S, beta - substable random vectors

An S α S random vector Xis β-substable, α < β ≤ 2, if Xd = Y Θ 1/β for some symmetric β-stable random vector Y Θ ≥ 0 a random variable with the Laplace transform exp{−tα/β}, Y and Θ are independent. We say that an S α S random vector is maximal if it is not β-substable for any β > α. In the paper we show that the ca,nonical spectral measure for every S α S, β-substablerandom vector X, β > α is equivalent to the Lebesgue measure on Sn−1.We show also that every such vector admits the representation X=Y+Z, where Y is an S α S sub-Gaussian random vector, Z is a maximal S α S random vector, Y and Z are independent. The last representation is not unique.