exchangeable random variables
A finite set^{} of random variables^{} $\{{X}_{1},\mathrm{\dots},{X}_{n}\}$ defined on a common probablility space $(\mathrm{\Omega},\mathcal{F},P)$ is said to be exchangeable if
$$P(({X}_{1}\in {B}_{1})\cap \mathrm{\cdots}\cap ({X}_{n}\in {B}_{n}))=P(({X}_{\sigma (1)}\in {B}_{1})\cap \mathrm{\cdots}\cap ({X}_{\sigma (n)}\in {B}_{n}))$$ 
for every set of Borel sets $\{{B}_{1},\mathrm{\dots},{B}_{n}\}$, and every permutation^{} $\sigma \in {S}_{n}$. In other words, ${X}_{1},\mathrm{\dots},{X}_{n}$ are exchangeable if their joint probability distribution function is the same regardless of their order.
A stochastic process^{} $\{{X}_{i}\}$ is said to be exchangeable if every finite subset of $\{{X}_{i}\}$ is exchangeable.
Remarks

•
If $S=\{{X}_{1},\mathrm{\dots},{X}_{n}\}$ is exchangeable, then every subset of $S$ is exchangeable (by picking suitable ${B}_{i}$ and $\sigma $). In particular, all ${X}_{i}$ are identically distributed, for
$$P({X}_{i}\in B)=P(({X}_{i}\in B)\cap ({X}_{j}\in \mathbb{R}))=P(({X}_{j}\in B)\cap ({X}_{i}\in \mathbb{R}))=P({X}_{j}\in B).$$ 
•
If $S=\{{X}_{1},\mathrm{\dots},{X}_{n}\}$ is iid, then $S$ is exchangeable, since the joint distribution^{} of ${X}_{i}$ is the product^{} of the distributions^{} of ${X}_{i}$:
$$P(({X}_{1}\in {B}_{1})\cap \mathrm{\cdots}\cap ({X}_{n}\in {B}_{n}))=P({X}_{\sigma (1)}\in {B}_{1}))\mathrm{\cdots}P({X}_{\sigma (n)}\in {B}_{n})).$$
Title  exchangeable random variables 

Canonical name  ExchangeableRandomVariables 
Date of creation  20130322 16:25:53 
Last modified on  20130322 16:25:53 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  5 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 60G09 
Synonym  exchangeable stochastic process 
Defines  exchangeable 
Defines  exchangeable process 