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

- Received:

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

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## Summary

Let *X, X*_{1},*X*_{2},... be i.i.d. *d*-dimensional random vectors with partial sums *S*_{n}. We identify the collection of random vectors *X* for which there exist non-singular linear operators *T*_{n} and vectors *υ*_{n}∈ℝ ^{d} such that {*ℒ(T*_{n}(*S*_{n}−υ_{n})),*n*>=1} is tight and has only full weak subsequential limits. The proof is constructive, providing a specific sequence {*T*_{n}}. 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 {*T*_{n}} and {*υ*_{n}} such that ℒ(*T*_{n}(*S*_{n}−υ_{n}))→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.