## The Annals of Probability

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

### Matrix Normalized Sums of Independent Identically Distributed Random Vectors

#### 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