The Annals of Probability

Approximate Independence of Distributions on Spheres and Their Stability Properties

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

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