Lacunary sequences and conditional independence (Q1911356)

The following interesting structural theorem is proved: let \(\{X_n\}\) be an arbitrary (not necessarily tight) sequence of real random variables (r.v.) with tail \(\sigma\)-field \(\mathcal I\); then, after a suitable enlargement of the basic probability space, one can find a subsequence \(\{X_{n_k}\}\) and a sequence of r.v. \(\{Y_k\}\) such that the \(Y_k\)'s are conditionally independent with respect to \(\mathcal I\) and \(\sum_{k \geq 1}|X_{n_k} - Y_k|< \infty\) a.s. The point made is that the conditional distributions of the \(Y_k\)'s given \(\mathcal I\) are not necessarily identical so that the sequence \(\{Y_k\}\) above is not necessarily an exchangeable sequence; an example given illustrates the point.