Affine normability of partial sums of I.I.D. random vectors: A characterization

  • Marjorie G. Hahn
  • Michael J. Klass
Article

Summary

Let X, X1,X2,... be i.i.d. d-dimensional random vectors with partial sums Sn. We identify the collection of random vectors X for which there exist non-singular linear operators Tn and vectors υn∈ℝ d such that {ℒ(Tn(Sn−υn)),n>=1} is tight and has only full weak subsequential limits. The proof is constructive, providing a specific sequence {Tn}. The random vector X is said to be in the generalized domain of attraction (GDOA) of a necessarily operator-stable law γ if there exist {Tn} and {υn} such that ℒ(Tn(Sn−υn))→γ. We characterize the GDOA of every operator-stable law, thereby extending previous results of Hahn and Klass; Hudson, Mason, and Veeh; and Jurek. The characterization assumes a particularly nice form in the case of a stable limit. When γ is symmetric stable, all marginals of X must be in the domain of attraction of a stable law. However, if γ is a nonsymmetric stable law then X may be in the GDOA of γ even if no marginal is in the domain of attraction of any law.

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References

  1. Billingsley, P.: Convergence of Probability Measures. New York: Wiley (1968)Google Scholar
  2. Hahn, M.G.: The generalized domain of attraction of a Gaussian law on Hilbert space. Lecture Notes in Math. 709, 125–144 (1979)Google Scholar
  3. Hahn, M.G., Hahn, P., Klass, M.J.: Pointwise translation of the Radon transform and the general central limit problem. Ann. Probability 11, 277–301 (1983)Google Scholar
  4. Hahn, M.G., Klass, M.J.: Matrix normalization of sums of random vectors in the domain of attraction of the multivariate normal. Ann. Probability 8, 262–280 (1980a)Google Scholar
  5. Hahn, M.G., Klass, M.J.: The generalized domain of attraction of spherically symmetric stable laws on ℝd Lecture Notes in Math. 828, 52–81 (1980b)Google Scholar
  6. Hahn, M.G., Klass, M.J.: The multidimensional central limit theorem for arrays normed by affine transformations. Ann. Probability 9, 611–623 (1981a)Google Scholar
  7. Hahn, M.G., Klass, M.J.: A survey of generalized domains of attraction and operator norming methods. Lecture Notes in Math. 860, 187–218 (1981b)Google Scholar
  8. Hudson, W.N.: Operator-stable distributions and stable marginals. J. Multivariate 10, 1–12 (1980)Google Scholar
  9. Hudson, W.N., Mason, J.D.: Operator-stable laws. J. Multivariate Anal. 11, 434–447 (1981)Google Scholar
  10. Hudson, W.N., Mason, J.D., Veeh, J.A.: The domain of normal attraction of an operator-stable law. Ann. Probability 11, 178–184 (1983)Google Scholar
  11. Jurek, Z.: Remarks on operator-stable probability measures. Comment. Math. XXI (1978)Google Scholar
  12. Jurek, Z.: Domains of normal attraction of operator-stable measures on Euclidean spaces. Bull. Acad. Polon. Sci. 28, 397–409 (1980)Google Scholar
  13. Kolmogorov, A.N.: On the approximation of distributions of sums of independent summands by infinitely divisible distributions. Sankhyā 25, 159–174 (1963)Google Scholar
  14. Le Cam, L.: On the distribution of sums of independent random variables. Bernoulli-BayesLaplace 179–202, Berlin Heidelberg New York: Springer (1965)Google Scholar
  15. Prohorov, Y.V.: On the uniform limit theorems of A.N. Kolmogorov. Teoria Veroy 5, 98–106 (1960)Google Scholar
  16. Sharpe, M.: Operator-stable probability measures on vector groups. Trans. Amer. Math. Soc. 136, 51–65 (1969)Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Marjorie G. Hahn
    • 1
  • Michael J. Klass
    • 2
  1. 1.MathematicsTufts UniversityMedfordUSA
  2. 2.University of CaliforniaBerkeley

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