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August, 1981 The Multidimensional Central Limit Theorem for Arrays Normed by Affine Transformations
Marjorie G. Hahn, Michael J. Klass
Ann. Probab. 9(4): 611-623 (August, 1981). DOI: 10.1214/aop/1176994366


Let $X_{n1}, \cdots, X_{nk_n}$ be independent random vectors in $\mathbb{R}^d$. Necessary and sufficient conditions are found for the existence of linear operators $A_n$ on $\mathbb{R}^d$ such that $\mathscr{L}(A_n(\sum^{k_n}_{j = 1} X_{nj})) \rightarrow N(\overset{\rightarrow}{0}, I)$, where $I$ is the $d \times d$ identity covariance matrix. These results extend the authors' previous work on sums of i.i.d. random vectors. The proof of the main theorem is constructive, yielding explicit centering vectors and norming linear operators.


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Marjorie G. Hahn. Michael J. Klass. "The Multidimensional Central Limit Theorem for Arrays Normed by Affine Transformations." Ann. Probab. 9 (4) 611 - 623, August, 1981.


Published: August, 1981
First available in Project Euclid: 19 April 2007

zbMATH: 0465.60030
MathSciNet: MR624687
Digital Object Identifier: 10.1214/aop/1176994366

Primary: 60F05

Keywords: central limit theorem , multivariate normal , operator normalization , translated trimming , triangular array

Rights: Copyright © 1981 Institute of Mathematical Statistics

Vol.9 • No. 4 • August, 1981
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