Invariance principles for generalized domains of semistable attraction

Let X,X1,X2,… be independent and identically distributed Rd-valued random vectors and assume X belongs to the generalized domain of attraction of some operator semistable law without normal component. Then without changing its distribution, one can redefine the sequence on a new probability space such that the properly affine normalized partial sums converge in probability and consequently even in Lp (for some p>0) to the corresponding operator semistable Lévy motion.