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 \(\nu (\omega )(B) = \mathbb {E}[\mathbf{1}_{X_0 \in B} \mid \mathcal{T}](\omega )\).