exchangeable random variables

A finite set of random variables $\{X_1,\mathrm{\dots },X_n\}$ defined on a common probablility space $(\mathrm{\Omega },ℱ,P)$ 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

  $$P(X_i\in B)=P((X_i\in B)\cap (X_j\in ℝ))=P((X_j\in B)\cap (X_i\in ℝ))=P(X_j\in B).$$

• 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 $X_i$:

  $$P((X_1\in B_1)\cap \mathrm{\cdots }\cap (X_n\in B_n))=P(X_{\sigma (1)}\in B_1))\mathrm{\cdots }P(X_{\sigma (n)}\in B_n)).$$