Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete

, Volume 69, Issue 4, pp 479–505

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

  • Marjorie G. Hahn
  • Michael J. Klass

DOI: 10.1007/BF00532663

Cite this article as:
Hahn, M.G. & Klass, M.J. Z. Wahrscheinlichkeitstheorie verw Gebiete (1985) 69: 479. doi:10.1007/BF00532663


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.

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