Exchangeable random variables and the subsequence principle

Call a sequence {Xn} of r.v.'s ε-exchangeable if on the same probability space there exists an exchangeable sequence {Yn} such that P(|Xn -Yn|≧ε)≦ε for all n. We prove that any tight sequence {Xn} defined on a rich enough probability space contains ε-exchangeable subsequences for every ε>0. The distribution of the approximating exchangeable sequences is also described in terms of {Xn}. Our results give a convenient way to prove limit theorems for subsequences of general r.v. sequences. In particular, they provide a simplified way to prove the subsequence theorems of Aldous [1] and lead also to various extensions.