The Annals of Probability

Matrix Normalized Sums of Independent Identically Distributed Random Vectors

Philip S. Griffin

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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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Ann. Probab., Volume 14, Number 1 (1986), 224-246.

First available in Project Euclid: 19 April 2007

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Primary: 60F05: Central limit and other weak theorems

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


Griffin, Philip S. Matrix Normalized Sums of Independent Identically Distributed Random Vectors. Ann. Probab. 14 (1986), no. 1, 224--246. doi:10.1214/aop/1176992624.

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