Operator Semi-Stable Distributions

Abstract Jajte introduced the operator semi-stable distributions on R n in [2] and proved an important fact: A full distribution μ is operator semi-stable, if and only if, there exist a number c(0 < c < 1), a vector h ∈ R n , and a nonsingular linear operator B in R n such that the formula μ c  = Bμ*δ(h) holds. In this paper, we make use of the eigenvalue of the matrix B to give a necessary and sufficient condition for ∫|x|≤1|x| r M(dx) < ∞, where M is the Levy measure of μ. Also, we use the symmetric group of μ to characterize the operators B in (1).