Dependencies

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 block average \(A_n = \frac{1}{n}\sum _{i=0}^{n-1} f(X_i)\) for bounded measurable \(f\).

A sequence \((X_n)_{n \in \mathbb {N}}\) is conditionally i.i.d. with respect to measure \(\mu \) if there exists a probability kernel \(\nu : \Omega \to \mathrm{Measure}(\alpha )\) such that for any strictly monotone \(k : \mathrm{Fin}(m) \to \mathbb {N}\), the joint distribution of \((X_{k(0)}, \ldots , X_{k(m-1)})\) equals the mixture \(\mu .\mathrm{bind}(\omega \mapsto \nu (\omega )^{\otimes m})\).

A sequence \((X_n)_{n \in \mathbb {N}}\) is contractable with respect to measure \(\mu \) if for all \(m \in \mathbb {N}\) and all strictly increasing functions \(k, k' : \mathrm{Fin}(m) \to \mathbb {N}\), the joint distribution of \((X_{k(0)}, \ldots , X_{k(m-1)})\) equals that of \((X_{k'(0)}, \ldots , X_{k'(m-1)})\).

A sequence \((X_n)_{n \in \mathbb {N}}\) of random variables is exchangeable with respect to measure \(\mu \) if for every \(n \in \mathbb {N}\) and every permutation \(\sigma \) of \(\{ 0, \ldots , n-1\} \), the joint distribution of \((X_{\sigma (0)}, \ldots , X_{\sigma (n-1)})\) equals that of \((X_0, \ldots , X_{n-1})\).

The future filtration at level \(m\) is \(\mathcal{F}_m = \sigma (X_{m+1}, X_{m+2}, \ldots )\).

The shift-invariant \(\sigma \)-algebra consists of sets \(S\) such that \(\theta ^{-1}(S) = S\) where \(\theta \) is the shift operator.

The tail \(\sigma \)-algebra of a sequence \((X_n)\) is \(\mathcal{T} = \bigcap _{n=0}^{\infty } \sigma (X_n, X_{n+1}, \ldots )\).

Coordinates are conditionally independent given the tail \(\sigma \)-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 \(n \to \infty \).

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. \(\omega \).

Block averages converge in L\(^1\) to the conditional expectation given the tail.

Block averages converge in L\(^2\) to the conditional expectation given the tail.

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 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.

For exchangeable sequences, the conditional expectation of a product does not depend on the lag between coordinates.

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, the covariance \(\mathrm{Cov}(f(X_i), f(X_j))\) is constant for \(i \ne j\).

The Stieltjes measure constructed from the limiting CDF is a probability measure a.e.

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 \(B\).

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.

For finite products of indicators, the conditional expectation given \(\mathcal{F}_m\) factors as a product of individual conditional expectations.

The future filtration is antitone: \(m \le n\) implies \(\mathcal{F}_n \le \mathcal{F}_m\).

For contractable sequences, indicator products satisfy the bridge condition.

For contractable sequences, integrals of products factor through block averages.

Conditional expectations of indicators given the reverse filtration converge to conditional expectations given the tail \(\sigma \)-algebra.

The key L\(^2\) bound that drives the Cesaro convergence.

The conditional expectation operator onto the shift-invariant subspace commutes with the Koopman operator.

For contractable sequences, certain L\(^2\) norms of block averages are bounded.

The L\(^2\) norm of the difference between averages over disjoint windows is bounded.

For a contractable sequence and \(k \le m\), the joint distribution of \((X_k, X_{m+1}, X_{m+2}, \ldots )\) equals that of \((X_m, X_{m+1}, X_{m+2}, \ldots )\).

Any strictly increasing function \(k : \mathrm{Fin}(m) \to \mathbb {N}\) with range contained in \(\{ 0, \ldots , n-1\} \) extends to a permutation of \(\{ 0, \ldots , n-1\} \).

Measures on product spaces are determined by their finite-dimensional marginals.

Products of block averages converge in L\(^1\).

The tail \(\sigma \)-algebra factorization follows from the finite-level factorization via reverse martingale convergence.

\(\mathcal{T} = \bigwedge _{m=0}^{\infty } \mathcal{F}_m\).

For all \(m\): \(\mathcal{T} \le \mathcal{F}_m\).

Birkhoff averages converge in L\(^2\) to the conditional expectation given the shift-invariant \(\sigma \)-algebra.

For contractable sequences, the conditional expectation of a product of indicators factors as a product of conditional expectations.

Conditional independence on indicators extends to the full product \(\sigma \)-algebra.

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\(^2\).

If \((X_n)\) is contractable, then it is conditionally i.i.d.

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.

If \((X_n)\) is exchangeable, then it is contractable.

For an infinite sequence \((X_n)_{n \in \mathbb {N}}\) of random variables taking values in a standard Borel space \(\alpha \) (with \(\alpha \) nonempty), the following are equivalent:

1. \((X_n)\) is contractable

2. \((X_n)\) is exchangeable

3. \((X_n)\) is conditionally i.i.d. (i.e., there exists a directing kernel \(\nu \))

Remark: The martingale proof constructs \(\nu \) from the tail \(\sigma \)-algebra \(\mathcal{T}\) via \(\nu (\omega )(B) = \mathbb {E}[\mathbf{1}_{X_0 \in B} \mid \mathcal{T}](\omega )\).

If \((X_n)\) is conditionally i.i.d., then it is exchangeable.

The monotone class theorem allows extending from indicators to measurable functions.

The weighted sums of indicators converge in L\(^1\).