Exchangeable random variables and the subsequence principle
Call a sequence \(\{X_ n\}\) of r.v.'s \(\epsilon\)-exchangeable if on the same probability space there exists an exchangeable sequence \(\{Y_ n\}\) such that \(P(| X_ n-Y_ n| \geq \epsilon)\leq \epsilon\) for all n. We prove that any tight sequence \(\{X_ n\}\) defined on a rich enough probability space contains \(\epsilon\)-exchangeable subsequences for every \(\epsilon >0\). The distribution of the approximating exchangeable sequences is also described in terms of \(\{X_ n\}\). 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 \textit{D. J. Aldous} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 40, 59-82 (1977)] and lead also to various extensions.
\(\epsilon \) -exchangeable subsequences
scientific article; zbMATH DE number 3913351