Infinite exchangeability

An infinite sequence of random variables \((X_1, X_2, ..., X_n, ...)\) is infinitely exchangeable under probability measure \(\mathbb{P}\) if the joint distribution for each subsequence \((X_{n_1},...,X_{n_k})\) satisfies \[ (X_{n_1},...,X_{n_k}) =^d (X_{\tau(n_1)},...,X_{\tau(n_k)}) \]

for all permutations \(\tau\) over \([k]\).