Exchangeable random variables

Remarks

Equivalently, an infinite sequence is exchangeable if its joint law on $S^{\N}$ is invariant under every permutation of $\N$ that moves only finitely many indices. Example: if $X_1, X_2, \dots$ are i.i.d., the joint law of $(X_1, \dots, X_n)$ is the product $\mu^{\otimes n}$ of the common law $\mu$, and a product of equal factors does not change when the factors are permuted. Another example: in the Pólya urn model, the sequence of colours drawn is exchangeable but not independent. Non-example: let $Y$ be a Bernoulli variable with parameter $\frac{1}{2}$, and let $X_1 = Y$ and $X_2 = 2Y$. Then $P((X_1, X_2) = (1, 2)) = \frac{1}{2}$ but $P((X_2, X_1) = (1, 2)) = 0$, so the transposition of $1$ and $2$ changes the law. Each $X_i$ of an exchangeable sequence has the same law, and each pair $(X_i, X_j)$ with $i \ne j$ has the same joint law.