# exchangeable random variables

A finite set of random variables $\{X_{1},\ldots,X_{n}\}$ defined on a common probablility space $(\Omega,\mathcal{F},P)$ is said to be exchangeable if

 $P\big{(}(X_{1}\in B_{1})\cap\cdots\cap(X_{n}\in B_{n})\big{)}=P\big{(}(X_{% \sigma(1)}\in B_{1})\cap\cdots\cap(X_{\sigma(n)}\in B_{n})\big{)}$

for every set of Borel sets $\{B_{1},\ldots,B_{n}\}$, and every permutation $\sigma\in S_{n}$. In other words, $X_{1},\ldots,X_{n}$ 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 $S=\{X_{1},\ldots,X_{n}\}$ 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

 $P(X_{i}\in B)=P\big{(}(X_{i}\in B)\cap(X_{j}\in\mathbb{R})\big{)}=P\big{(}(X_{% j}\in B)\cap(X_{i}\in\mathbb{R})\big{)}=P(X_{j}\in B).$
• If $S=\{X_{1},\ldots,X_{n}\}$ is iid, then $S$ is exchangeable, since the joint distribution of $X_{i}$ is the product of the distributions of $X_{i}$:

 $P\big{(}(X_{1}\in B_{1})\cap\cdots\cap(X_{n}\in B_{n})\big{)}=P(X_{\sigma(1)}% \in B_{1})\big{)}\cdots P(X_{\sigma(n)}\in B_{n})\big{)}.$
Title exchangeable random variables ExchangeableRandomVariables 2013-03-22 16:25:53 2013-03-22 16:25:53 CWoo (3771) CWoo (3771) 5 CWoo (3771) Definition msc 60G09 exchangeable stochastic process exchangeable exchangeable process