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Approximate independence of distributions on spheres and their stability properties. (English) Zbl 0732.62014
The authors prove the following result: Suppose $$\zeta_ i$$, $$i\geq 1$$, is a sequence of independent, identically distributed positive random variables with d.f. F satisfying the normalization E $$\zeta {}_ i^ p=1$$ for some $$0<p<\infty$$. Define $X_{k,n,p}=(\sum^{k}_{j=1}\zeta^ p_ j)/(\sum^{n}_{j=1}\zeta^ p_ j)$ for $$1\leq k\leq n$$, $$n\geq 1$$. There exists a unique distribution $$F_ p$$ with density $f_ p(x)=(p^{1-p}/\Gamma (1/p))\exp \{-x^ p/p\}$ such that $$X_{k,n,p}$$ has beta-distribution with parameters (k/p, (n-k)/p) for every $$k\leq n$$, $$n\geq 1$$. Define $X_{k,n,\infty}=\max (\zeta_ 1,...,\zeta_ k)/\max (\zeta_ 1,...,\zeta_ n)$ and let $$\gamma_{k,n,\infty}$$ have the distribution function $P(\gamma_{k,n,\infty}\leq x)=(n-k)n^{-1}x^ k\text{ for } 0\leq x<1,\text{ and } =1\text{ for } x\geq 1.$ Then $$X_{k,n,\infty}$$ and $$\gamma_{k,n,\infty}$$ are identically distributed for any $$k\leq n$$, $$n\geq 1$$, iff $$\zeta_ 1$$ has uniform distribution on [0,1]. Related characterization theorems and stability properties are discussed. Some discrete versions of the problem and applications to de Finetti-type theorems are investigated.

##### MSC:
 62E10 Characterization and structure theory of statistical distributions 62B99 Sufficiency and information 60E05 Probability distributions: general theory
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