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Approximate Independence of Distributions on Spheres and Their Stability Properties

S. T. Rachev and L. Ruschendorf
The Annals of Probability
Vol. 19, No. 3 (Jul., 1991), pp. 1311-1337
Stable URL: http://www.jstor.org/stable/2244484
Page Count: 27
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Approximate Independence of Distributions on Spheres and Their Stability Properties
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Abstract

Let ζ be chosen at random on the surface of the p-sphere in Rn, 0p,n := {x ∈ Rn: ∑n i = 1|xi|p = n}. If p = 2, then the first k components ζ1,..., ζk are, for k fixed, in the limit as n→∞ independent standard normal. Considering the general case $p > 0$, the same phenomenon appears with a distribution Fp in an exponential class E. 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.

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