Rates of convergence in the operator-stable limit theorem

Suppose that the ℝ^d-valued random vector 0 is strictly operator-stable in the sense that μ̂, the characteristic function of 0, satisfies μ̂(z)^t = μ̂(t^B*z) for every t > 0, for some invertible linear operator B on ℝ^d. Suppose also that for the i.i.d. random vectors {X_i} in ℝ^d, n^-B Σ_i=1ⁿ X_i→^w 0. In the present paper, we study the rates of convergence of this operator-stable limit theorem in terms of several probability metrics. A new type of "ideal" metrics suitable for this rate-of-convergence problem is introduced.