TITLE: What makes random variables exchangeable and what is implied by exchangeability?
QUESTION [20 upvotes]: Consider a $d$-dimensional random vector$\ X=(X_j)$. $\ X$ is called exchangeable if $\ (X_1,\ldots,X_d)\mathrel{\overset d =} ({X_{{j_1}}},\ldots,X_{j_d})$ for any permutation$\ j_1,\ldots, j_d$. If$\ X_j$ are iid,$\ X$ is exchangeable. The converse is false (correct me if I'm wrong). What can we say about just identically distributed $\ X_j$? What can we say about $\ X_j$ if we know that$\ X$ is exchangeable?

REPLY [14 votes]: A simple and rather general way to build an exchangeable random vector $X=(X_k)_{1\leqslant k\leqslant n}$ is to start from an i.i.d. sequence $(Y_k)_{1\leqslant k\leqslant n}$ and to define, for every $k$, $X_k=\Phi(U,Y_k)$, where $U$ is a random variable independent on $(Y_k)_{1\leqslant k\leqslant n}$ and $\Phi$ a measurable function.
And, in some (loose) sense, this is the only way... as shown by Diaconis and Freedman for finite sequences, using mixtures of urn sequences rather than mixtures of i.i.d. sequences, and earlier, by de Finetti for infinite sequences.