## The Annals of Probability

- Ann. Probab.
- Volume 19, Number 3 (1991), 1311-1337.

### Approximate Independence of Distributions on Spheres and Their Stability Properties

S. T. Rachev and L. Ruschendorf

#### Abstract

Let $\zeta$ be chosen at random on the surface of the $p$-sphere in $\mathbb{R}^n, 0_{p,n} := \{x \in \mathbb{R}^n: \sum^n_{i = 1}|x_i|^p = n\}$. If $p = 2$, then the first $k$ components $\zeta_1,\ldots, \zeta_k$ are, for $k$ fixed, in the limit as $n\rightarrow\infty$ independent standard normal. Considering the general case $p > 0$, the same phenomenon appears with a distribution $F_p$ in an exponential class $\mathscr{E}. F_p$ can be characterized by the distribution of quotients of sums, by conditional distributions and by a maximum entropy condition. These characterizations have some interesting stability properties. Some discrete versions of this problem and some applications to de Finetti-type theorems are discussed.

#### Article information

**Source**

Ann. Probab., Volume 19, Number 3 (1991), 1311-1337.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176990346

**Digital Object Identifier**

doi:10.1214/aop/1176990346

**Mathematical Reviews number (MathSciNet)**

MR1112418

**Zentralblatt MATH identifier**

0732.62014

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60E05: Distributions: general theory

Secondary: 62B20

**Keywords**

de Finetti's theorem characterization of distributions stability

#### Citation

Rachev, S. T.; Ruschendorf, L. Approximate Independence of Distributions on Spheres and Their Stability Properties. Ann. Probab. 19 (1991), no. 3, 1311--1337. doi:10.1214/aop/1176990346. https://projecteuclid.org/euclid.aop/1176990346