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.
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This paper was presented in the special session on Asymptotic Behavior of Sums at the Annual IMS Meeting in Cincinnati, August 18, 1982
This research was supported in part by National Science Foundation Grants MCS-81-01895 and MCS-83-01326
This research was supported in part by National Science Foundation Grants MCS-80-64022 and MCS-83-01793
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Hahn, M.G., Klass, M.J. Affine normability of partial sums of I.I.D. random vectors: A characterization. Z. Wahrscheinlichkeitstheorie verw Gebiete 69, 479–505 (1985). https://doi.org/10.1007/BF00532663
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DOI: https://doi.org/10.1007/BF00532663