Approximate independence of distributions on spheres and their stability properties
type: Item
instance of: scholarly article
title: Approximate independence of distributions on spheres and their stability properties
zbMATH Open document ID: 0732.62014
published in: The Annals of Probability
publication date: 1991-01-01T00:00:00Z
review text: 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.
reviewed by: B. L. S. Prakasa Rao
Mathematics Subject Classification ID: 62E10
Mathematics Subject Classification ID: 62B99
Mathematics Subject Classification ID: 60E05
zbMATH DE Number: 4211258
zbMATH Keywords: approximate independence
zbMATH Keywords: distributions on spheres
zbMATH Keywords: exponential class
zbMATH Keywords: distributions of quotients of sums
zbMATH Keywords: conditional distributions
zbMATH Keywords: maximum entropy condition
zbMATH Keywords: beta-distribution
zbMATH Keywords: characterization theorems
zbMATH Keywords: stability properties
zbMATH Keywords: discrete versions
zbMATH Keywords: Finetti-type theorems
author: Svetlozar T. Rachev
author: Ludger Rüschendorf
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176990346
OpenAlex ID: W2091661482
DOI: 10.1214/AOP/1176990346
description: scientific article; zbMATH DE number 4211258