A dozen de Finetti-style results in search of a theory
type: Item
instance of: scholarly article
title: A dozen de Finetti-style results in search of a theory
zbMATH Open document ID: 0619.60039
author: Persi Diaconis
author: David Freedman
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 1987-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77315
review text: It is shown that if \(\xi =(\xi_ 1,\xi_ 2,...,\xi_ n)\) is uniformly distributed on the surface of the sphere \(\{\xi\) :\(\sum^{n}_{i=1}\xi^ 2_ i=n\}\) and \(1\leq k\leq n-4\), then the variation distance between the law of \((\xi_ 1,...,\xi_ k)\) and the joint law of k independent standard normal variables is less than or equal to \(2(k+3)/(n-k-3)\). It follows from this that if a law in \({\mathbb{R}}^ k\) is orthogonally invariant, then it is within variation distance \(2(k+3)/(n-k-3)\) of a mixture of joint laws of i.i.d. centred normals. Similar results are proved for the exponential, geometric and Poisson distributions: for example if \((\xi_ 1,...,\xi_ n)\) is uniform on the simplex \(\{\xi:\xi_ i\geq 0\) for all i and \(\sum^{n}_{i=1}\xi_ i=n\}\) then, for \(1\leq k\leq n-2\), the law of \((\xi_ 1,...,\xi_ k)\) is within variation distance \(2(k+1)/(n-k+1)\) of the joint law of k i.i.d. exponential variables with parameter 1. The paper discusses sharpness of bounds, questions of uniqueness of mixtures etc., and concludes with extensive historical remarks.
Mathematics Subject Classification ID: 60G09
Mathematics Subject Classification ID: 60J05
Mathematics Subject Classification ID: 60G10
zbMATH DE Number: 4005251
zbMATH Keywords: de Finetti's theorem
zbMATH Keywords: exchangeable
zbMATH Keywords: orthogonally invariant
zbMATH Keywords: variation distance
zbMATH Keywords: uniqueness of mixtures
zbMATH Keywords: historical remarks
reviewed by: F. Papangelou
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4005251