Class L of multivariate distributions and its subclasses

For any class Q of distributions on Rd, let (Q) be the class of limit distributions of bn-1(X1 + ... + Xn) - an, where {Xn} are independent Rd-valued random variables, each with distribution in Q, bn > 0, an [set membership, variant] Rd, and {bn-1Xj} is a null array. When Q is the class of all distributions on Rd. (Q) = L0 is the usual class L. Define Lm = (Lm-1) and L[infinity] = [intersection]Lm. It is shown that this definition of Lm is equivalent to Urbanik's definition. Description of Lévy measures and representation of characteristic functions of members in these classes are given. Other characterizations of the class L[infinity] are made. Conditions for convergence in terms of the representations are given. Continuity properties of distributions of class L are studied.