This article contains a proof of the conjectured central limit theorem for the growth of the right-hand edge for supercritical oriented percolation. The technique of proof, finding points with regeneration-type properties termed "break points," may also apply to other processes.
"The Central Limit Theorem for the Right Edge of Supercritical Oriented Percolation." Ann. Probab. 17 (4) 1322 - 1332, October, 1989. https://doi.org/10.1214/aop/1176991157