We consider self-averaging sequences in which each term is a weighted average over previous terms. For several sequences of this kind it is known that they do not converge to a limit. These sequences share the property that $n$th term is mainly based on terms around a fixed fraction of $n$. We give a probabilistic interpretation to such sequences and give weak conditions under which it is natural to expect non-convergence. Our methods are illustrated by application to the group Russian roulette problem.
"Self-averaging sequences which fail to converge." Electron. Commun. Probab. 22 1 - 12, 2017. https://doi.org/10.1214/17-ECP48