## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 28, Number 4 (1957), 978-986.

### Random Unit Vectors II: Usefulness of Gram-Charlier and Related Series in Approximating Distributions

David Durand and J. Arthur Greenwood

#### Abstract

The distribution of the sum of $n$ random coplanar unit vectors and of a given component of the sum has been discussed by many authors, who have shown that each distribution can be approximated in series that are asymptotically normal. But the difficult question of the usefulness of these approximations for finite $n$--in particular for small $n$--has not been exhaustively treated. Accordingly, this paper reexamines some analyses of Pearson's series for the vector sum, presents corresponding series for a component, and examines the accuracy of the latter series.

#### Article information

**Source**

Ann. Math. Statist., Volume 28, Number 4 (1957), 978-986.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177706798

**Digital Object Identifier**

doi:10.1214/aoms/1177706798

**Mathematical Reviews number (MathSciNet)**

MR93805

**Zentralblatt MATH identifier**

0083.14005

**JSTOR**

links.jstor.org

#### Citation

Durand, David; Greenwood, J. Arthur. Random Unit Vectors II: Usefulness of Gram-Charlier and Related Series in Approximating Distributions. Ann. Math. Statist. 28 (1957), no. 4, 978--986. doi:10.1214/aoms/1177706798. https://projecteuclid.org/euclid.aoms/1177706798

#### See also

- Part I: J. Arthur Greenwood, David Durand. The Distribution of Length and Components of the Sum of $n$ Random Unit Vectors. Ann. Math. Statist., Volume 26, Number 2 (1955), 233--246.Project Euclid: euclid.aoms/1177728540