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

Abstract

Let X, X1,X2,... 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 υn∈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.