Backwards Martingales and Exchangeability

Abstract

With many data acquisitions, such as telephone surveys, the order in which the data come does not matter. Mathematically, we say that a family of random variables is exchangeable if the joint distribution does not change under finite permutations. De Finetti’s structural theorem says that an infinite family of E-valued exchangeable random variables can be described by a two-stage experiment. At the first stage, a probability distribution Ξ on E is drawn at random. At the second stage, independent and identically distributed random variables with distribution Ξ are implemented.

We first define the notion of exchangeability. Then we consider backwards martingales and prove the convergence theorem for them. This is the cornerstone for the proof of de Finetti’s theorem.