Approximate Independence of Distributions on Spheres and Their Stability Properties

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.