## The Annals of Probability

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

#### 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

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