The Annals of Probability

Matrix Normalized Sums of Independent Identically Distributed Random Vectors

Philip S. Griffin

Full-text: Open access

Abstract

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.

Article information

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

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176992624

Digital Object Identifier
doi:10.1214/aop/1176992624

Mathematical Reviews number (MathSciNet)
MR815967

Zentralblatt MATH identifier
0602.60031

JSTOR
links.jstor.org

Subjects
Primary: 60F05: Central limit and other weak theorems

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

Citation

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. https://projecteuclid.org/euclid.aop/1176992624


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