Optional skipping of martingale differences and related sequences (Q1374096)

Let \((w_i)_{i\geq 1}\) be a sequence of random variables adapted to the filtration \((G_i)_{i\geq 1}\). Let \((\tau(j))_{j\geq 1}\) be a strictly increasing sequence of almost surely finite random variables which are predetermined with respect to \((G_i)_{i\geq 1}\). Define the optionally skipped sequence \((\nu_j)_{j\geq 1}\) by \(\nu_j= w_{\tau(j)}\). This paper establishes relationships between conditional expectations of the optionally skipped variables and conditional expectations of the original sequence. These results generalize similar results that have been established for the special case where the \(w_i\) are independent and identically distributed and where \((w_i)\) is a martingale difference sequence.