Summary
If random variables X n converge in distribution to a nondegenerate random variable X, and if transformations a n X n +b n of them, with a n >0, also converge in distribution to X, then a n →1 and b n →0. Moreover, the group of transformations x→ax+b that preserve the distribution of a nondegenerate random variable consists either of the identity alone, or of the identity together with the reflection through some point. For proofs see [4], [5], or [6]. This paper gives the corresponding results for k-dimensional random vectors.
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This research was supported in part by Research Grant No. NSF-GP 3707 from the Division of Mathematical, Physical, and Engineering Sciences of the National Science Foundation.
I should like thank Hans BrØns and SØren Johansen for very helpful discussions.
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Billingsley, P. Convergence of types in k-space. Z. Wahrscheinlichkeitstheorie verw Gebiete 5, 175–179 (1966). https://doi.org/10.1007/BF00536653
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DOI: https://doi.org/10.1007/BF00536653