De Finetti theorem
= De Finetti theorem
{c}
= Conditional independence of exchangeable sequences
{synonym}
For an infinite sequence of real <exchangeable random variables>, conditional on its <invariant sigma-algebra of an exchangeable sequence>, the coordinates are independent and identically distributed. In particular, for Borel sets $A_0,\ldots,A_p$,
$$
\mathbb E\!\left[\prod_{k=0}^p\mathbf1_{A_k}(X_k)\mid\mathcal S\right]
=\prod_{k=0}^p\mathbb E[\mathbf1_{A_k}(X_k)\mid\mathcal S].
$$
Here is the core proof. Let $\mathcal S_m$ denote invariance under permutations of coordinates $0,\ldots,m$. <Symmetrization as conditional expectation> identifies the empirical average of each indicator with its conditional expectation on $\mathcal S_m$. The <reverse martingale convergence theorem> gives its limit conditional on $\mathcal S$. Multiplying the averages sums over all ordered coordinate tuples, including repetitions. Symmetrization of the product itself sums over the distinct tuples. The fraction of repeated tuples is at most $\binom{p+1}{2}/(m+1)$, so these bounded averages have the same limit. This proves the displayed factorization. Permutation invariance also makes the individual conditional laws identical. A regular conditional distribution on the real line packages that common law as a random probability measure, giving the equivalent mixture-of-iid formulation.