Abstract
Suppose that the ℝd-valued random vector θ is strictly operator-stable in the sense that\(\hat \mu \), the characteristic function of θ, satisfies\(\hat \mu (z)^t = \hat \mu (t^{B*} z)\) for everyt<0, for some invertible linear operatorB on ℝd. Suppose also that for the i.i.d. random vectors {X i } in ℝd,\(n^{ - B} \Sigma _{i = 1}^n X_i \xrightarrow{w}\theta \). 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.
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This research was partially supported by NSF, Grant DMS-9103452 and NATO, Grant CRG900798.
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Maejima, M., Rachev, S.T. Rates of convergence in the operator-stable limit theorem. J Theor Probab 9, 37–85 (1996). https://doi.org/10.1007/BF02213734
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DOI: https://doi.org/10.1007/BF02213734