## The Annals of Probability

- Ann. Probab.
- Volume 8, Number 2 (1980), 262-280.

### Matrix Normalization of Sums of Random Vectors in the Domain of Attraction of the Multivariate Normal

Marjorie G. Hahn and Michael J. Klass

#### Abstract

Let $S_n$ be a sequence of partial sums of mean zero purely $d$-dimensional i.i.d. random vectors. Necessary and sufficient conditions are given for the existence of matrices $A_n$ such that the transform of $S_n$ by $A_n$ is asymptotically multivariate normal with identity covariance matrix. This is more general than previous $d$-dimensional results. Examples are given to illustrate the need for the present approach. The matrices $A_n$ take a particularly simple form because of a degree of uncorrelatedness between certain pairs of 1-dimensional random variables obtained by projection.

#### Article information

**Source**

Ann. Probab., Volume 8, Number 2 (1980), 262-280.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176994776

**Digital Object Identifier**

doi:10.1214/aop/1176994776

**Mathematical Reviews number (MathSciNet)**

MR566593

**Zentralblatt MATH identifier**

0428.60032

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

**Keywords**

Central limit theorem truncated correlation infinite variance matrix normalization multivariate normal random vectors

#### Citation

Hahn, Marjorie G.; Klass, Michael J. Matrix Normalization of Sums of Random Vectors in the Domain of Attraction of the Multivariate Normal. Ann. Probab. 8 (1980), no. 2, 262--280. doi:10.1214/aop/1176994776. https://projecteuclid.org/euclid.aop/1176994776