Exchangeability and semigroups

Abstract

Exchangeability of a “random object” is a strong symmetry condition, leading in general to a convex set of distributions not too far from a “simplex” - a set easily described by its extreme points, in this case distributions with very special properties as for example iid coin tossing sequences in de Finetti’s original result. Although in most cases of interest the symmetry is defined via a non–commutative group acting on the underlying space, it very often can be described by a suitable factorization involving an abelian semigroup. The factorizing function typically turns out to be positive definite, and results from Harmonic Analysis on semigroups become applicable. In this way many known theorems on exchangeability can be given an alternative proof, more analytic/algebraic in a sense, but also new results become available.

Keywords

• exchangeability
• semigroups
• positive definite functions
• de finetti’s theorem
• random partition