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Norming operators for generalized domains of attraction

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Abstract

A sequence of independent and identically distributed random vectorsXn on Rk is said to belong to the generalized domain of attraction of a nondegenerate random vectorY on Rk provided that there exist linear operatorsAn on Rk and nonrandom constantsbn ? Rk such that the centered and normalized partial sumsAn(X1+⋯+Xn−bn converge in distribution toY. In this paper we show that the sequence of norming operatorsAn can always be chosen to vary regularly.

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References

  1. Billingsley, P. (1966). Convergence of types ink-space.Z. Wahrsch. verw. Geb. 5, 175–179.

    Article  Google Scholar 

  2. Hahn, M., and Klass, M. (1980). Matrix normalization for sums of random vectors in the domain of attraction of the multivariate normal.Ann. Prob. 8, 262–280.

    Article  MathSciNet  Google Scholar 

  3. Holmes, J., Hudson, W., and Mason, J. D. (1982). Operator stable laws: multiple exponents and elliptical symmetry.Ann. Prob. 10, 602–612.

    Article  MathSciNet  Google Scholar 

  4. Hudson, W., Jurek, Z., and Veeh, J. (1986). The symmetry group and exponents of operator stable probability measures.Ann. Prob. 14, 1014–1023.

    Article  MathSciNet  Google Scholar 

  5. Michaliček, J. (1972). Der Anziehungsbereich von Operator-Stabilen Verteilungen imR2.Z. Wahrsch. verw. Geb. 25, 57–70.

    Article  MathSciNet  Google Scholar 

  6. Meerschaert, M. (1988). Regular variation in Rk.Proc. Amer. Math. Soc. 102, 341–348.

    MathSciNet  MATH  Google Scholar 

  7. Meerschaert, M. (1991). Spectral decomposition for generalized domains of attraction.Ann. Prob. 19, 875–892.

    Article  MathSciNet  Google Scholar 

  8. Sharpe, M. (1969). Operator-stable probability distributions on vector groups.Trans. Amer. Math. Soc. 136, 51–65.

    Article  MathSciNet  Google Scholar 

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Author notes

  1. Mark M. Meerschaert

    Present address: Department of Mathematics, Albion College, 49224, Albion, Michigan

Authors and Affiliations

  1. Department of Statistics and Probability, Michigan State University, 48824, East Lansing, Michigan

    Mark M. Meerschaert

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  1. Mark M. Meerschaert
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Additional information

Partially supported by NSF Grant DMS-91-03131 at Albion College.

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Meerschaert, M.M. Norming operators for generalized domains of attraction. J Theor Probab 7, 793–798 (1994). https://doi.org/10.1007/BF02214372

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  • DOI: https://doi.org/10.1007/BF02214372

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