On a representation theorem for finitely exchangeable random vectors

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:442

Article

Janson, Svante¹  Konstantopoulos, Takis¹  Yuan, Linglong^2,3 

[1] Uppsala Univ, Dept Math, POB 480, SE-75106 Uppsala, Sweden

[2] Johannes Gutenberg Univ Mainz, Inst Math, Staudingerweg 9, D-55099 Mainz, Germany

[3] Uppsala Univ, SE-75106 Uppsala, Sweden

关键词: Signed measure;    Measurable space;    Point measure;    Exchangeable;    Symmetric;    Homogeneous polynomial;   

DOI  :  10.1016/j.jmaa.2016.04.070

来源: Elsevier

【 摘 要 】

A random vector X = (X1, ... ,X-n) with the X-i taking values in an arbitrary measurable space (S, Sp) is exchangeable if its law is the same as that of (X-sigma(1), ... ,X-sigma(n)) for any permutation a. We give an alternative and shorter proof of the representation result (Jaynes [6] and Kerns and Szekely [9]) stating that the law of X is a mixture of product probability measures with respect to a signed mixing measure. The result is finitistic in nature meaning that it is a matter of linear algebra for finite S. The passing from finite S to an arbitrary one may pose some measure-theoretic difficulties which are avoided by our proof. The mixing signed measure is not unique (examples are given), but we pay more attention to the one constructed in the proof (canonical mixing measure) by pointing out some of its characteristics. The mixing measure is, in general, defined on the space of probability measures on S; but for S =, one can choose a mixing measure on R-n. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free