Approximate Independence of Distributions on Spheres and Their Stability Properties

Let ; be chosen at random on the surface of the p-sphere in R , Op n:"= {x E Rn: E 1xix1P = n}. If p = 2, then the first k components ;1, .k are, for k fixed, in the limit as n -X oo independent standard normal. Considering the general case p > 0, the same phenomenon appears with a distribution Fp in an exponential class 6'. Fp 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.