Approximate independence of distributions on spheres and their stability properties. (English) Zbl 0732.62014

Rachev, S. T.; Rüschendorf, L.

Ann. Probab. 19, No. 3, 1311-1337 (1991).

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.

Reviewer: B.L.S.Prakasa Rao (New Delhi)

MSC:

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

Keywords:

approximate independence; distributions on spheres; exponential class; distributions of quotients of sums; conditional distributions; maximum entropy condition; beta-distribution; characterization theorems; stability properties; discrete versions; Finetti-type theorems