4 Main Implication: Contractable implies Conditionally i.i.d.
This is the deep direction of de Finetti’s theorem. We formalize three independent proofs.
4.1 Via Martingale (Aldous’ proof)
The martingale approach uses reverse martingale convergence to the tail \(\sigma \)-algebra.
4.1.1 Pair Law Equality
For a contractable sequence and \(k \le m\), the joint distribution of \((X_k, X_{m+1}, X_{m+2}, \ldots )\) equals that of \((X_m, X_{m+1}, X_{m+2}, \ldots )\).
For a contractable sequence, the joint distribution of \((X_k, \theta _{m+1} X)\) equals that of \((X_m, \theta _{m+1} X)\) for all \(k \le m\), where \(\theta _n\) is the shift operator.
For contractable sequences and \(k \le m\): \(\mathbb {E}[\mathbf{1}_{X_k \in B} \mid \mathcal{F}_m] = \mathbb {E}[\mathbf{1}_{X_m \in B} \mid \mathcal{F}_m]\) a.s.
4.1.2 Kallenberg Chain and Convergence
Conditional expectations of indicators given the reverse filtration converge to conditional expectations given the tail \(\sigma \)-algebra.
For \(k \le m\) and measurable \(B\): \(\mathbb {E}[\mathbf{1}_{X_m \in B} \mid \mathcal{F}_m] = \mathbb {E}[\mathbf{1}_{X_k \in B} \mid \mathcal{F}_m]\) a.s.
For any measurable \(B\): \(\mathbb {E}[\mathbf{1}_{X_m \in B} \mid \mathcal{T}] = \mathbb {E}[\mathbf{1}_{X_0 \in B} \mid \mathcal{T}]\) a.s.
4.1.3 Factorization and Directing Measure
For finite products of indicators, the conditional expectation given \(\mathcal{F}_m\) factors as a product of individual conditional expectations.
The tail \(\sigma \)-algebra factorization follows from the finite-level factorization via reverse martingale convergence.
Coordinates are conditionally independent given the tail \(\sigma \)-algebra.
The directing measure \(\nu (\omega )(B) = \mathbb {E}[\mathbf{1}_{X_0 \in B} \mid \mathcal{T}](\omega )\) is a probability measure for a.e. \(\omega \).
The directing measure \(\omega \mapsto \nu (\omega )(B)\) is measurable for each Borel \(B\).
If \((X_n)\) is contractable, then it is conditionally i.i.d. The directing kernel \(\nu (\omega )(B) = \mathbb {E}[\mathbf{1}_{X_0 \in B} \mid \mathcal{T}](\omega )\) is constructed from the tail \(\sigma \)-algebra.
4.2 Via L\(^2\) (Elementary proof)
The L\(^2\) approach uses elementary contractability bounds on block averages. This is Kallenberg’s “second proof” and has the lightest dependencies.
Note: This proof applies to real-valued sequences (\(X : \mathbb {N} \to \Omega \to \mathbb {R}\)) with \(L^2\) integrability (i.e., \(\mathbb {E}[X_i^2] {\lt} \infty \) for all \(i\)).
4.2.1 Block Averages and Covariance Structure
The block average \(A_n = \frac{1}{n}\sum _{i=0}^{n-1} f(X_i)\) for bounded measurable \(f\).
For contractable sequences, the covariance \(\mathrm{Cov}(f(X_i), f(X_j))\) is constant for \(i \ne j\).
The L\(^2\) norm of the difference between averages over disjoint windows is bounded.
For contractable sequences, certain L\(^2\) norms of block averages are bounded.
4.2.2 Cesaro Convergence
The key L\(^2\) bound that drives the Cesaro convergence.
Block averages converge in L\(^2\) to the conditional expectation given the tail.
Block averages converge in L\(^1\) to the conditional expectation given the tail.
4.2.3 Directing Measure Construction
The limiting function \(\alpha (t, \omega )\) is a.e. in \([0,1]\).
The limiting function \(\alpha (t, \omega )\) is monotone in \(t\) for a.e. \(\omega \).
The Stieltjes measure constructed from the limiting CDF is a probability measure a.e.
The weighted sums of indicators converge in L\(^1\).
If \((X_n)\) is contractable, then it is conditionally i.i.d.
4.3 Via Koopman (Mean Ergodic Theorem)
The Koopman approach uses the Mean Ergodic Theorem via the shift operator on L\(^2\). This is Kallenberg’s “first proof” and uses disjoint-block averaging.
4.3.1 Block Averages and Ergodic Theory
The block average \(A_{m,n,k}(f)(\omega ) = \frac{1}{n} \sum _{j=0}^{n-1} f(\omega _{k \cdot n + j})\) averages \(f\) over the \(k\)-th block of size \(n\) (indices \([kn, kn+n)\)). For \(n = 0\), the block average is defined as \(0\).
The conditional expectation operator onto the shift-invariant subspace commutes with the Koopman operator.
Birkhoff averages converge in L\(^2\) to the conditional expectation given the shift-invariant \(\sigma \)-algebra.
For a shift-invariant measure, block averages converge in \(L^1\) to the conditional expectation given the shift-invariant \(\sigma \)-algebra: \(\int |A_{m,n,k}(f) - \mathbb {E}[f \circ \pi _0 \mid \mathcal{I}]| \, d\mu \to 0\) as \(n \to \infty \).
4.3.2 Contractability and Factorization
For exchangeable sequences, the conditional expectation of a product does not depend on the lag between coordinates.
For contractable sequences, integrals of products factor through block averages.
Products of block averages converge in L\(^1\).
For contractable sequences, the conditional expectation of a product of indicators factors as a product of conditional expectations.
For contractable sequences, indicator products satisfy the bridge condition.
If \((X_n)\) is contractable, then it is conditionally i.i.d. This proof uses the Mean Ergodic Theorem via the Koopman operator on L\(^2\).