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January, 1986 Matrix Normalized Sums of Independent Identically Distributed Random Vectors
Philip S. Griffin
Ann. Probab. 14(1): 224-246 (January, 1986). DOI: 10.1214/aop/1176992624


Let $X_1, X_2,\cdots$ be a sequence of independent identically distributed random vectors and $S_n = X_1 + \cdots + X_n$. Necessary and sufficient conditions are given for there to exist matrices $B_n$ and vectors $\gamma_n$ such that $\{B_n(S_n - \gamma_n)\}$ is stochastically compact, i.e., $\{B_n(S_n - \gamma_n)\}$ is tight and no subsequential limit is degenerate. When this condition holds we are able to obtain precise estimates on the distribution of $S_n$. These results are then specialized to the case where $X_1$ is in the generalized domain of attraction of an operator stable law and a local limit theorem is proved which generalizes the classical local limit theorem where the normalization is done by scalars.


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Philip S. Griffin. "Matrix Normalized Sums of Independent Identically Distributed Random Vectors." Ann. Probab. 14 (1) 224 - 246, January, 1986.


Published: January, 1986
First available in Project Euclid: 19 April 2007

zbMATH: 0602.60031
MathSciNet: MR815967
Digital Object Identifier: 10.1214/aop/1176992624

Primary: 60F05

Keywords: generalized domain of attraction , local limit theorem , Matrix normalization , probability estimates , stochastic compactness , tightness

Rights: Copyright © 1986 Institute of Mathematical Statistics


Vol.14 • No. 1 • January, 1986
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