We provide experimental data on the initiation of persistent fission chains obtained at different supercritical states, using the fast burst reactor Caliban. In many previous papers, theory has been compared mostly with initiation experiments at various superprompt critical states, whereas very few experimental data have been published on delayed supercritical states. To fill the lack of data, we have conducted three studies on the reactor at reactivities far below 0.7 $, which is one of the lowest states ever published for a similar assembly. We give a justification of the use of the gamma function to fit experimental results for the temporal distributions of waiting times and compare experiments with numerical simulations obtained with a punctual zero-dimensional Monte Carlo code and a punctual deterministic initiation code.