A general set of distribution-free conditions is described under which an i.i.d. sequence of random variables is preserved under optional skipping. This work is motivated by theorems of J. L. Doob and Z. Ignatov, unifying and extending aspects of both.
"Doob, Ignatov and optional skipping." Ann. Probab. 30 (4) 1933 - 1958, October 2002. https://doi.org/10.1214/aop/1039548377