Extension of the critical-window theorem to unbounded car-arrival distributions

Establish that the critical-window conclusion of Theorem transition remains valid for unbounded car-arrival distributions, including Poisson and geometric distributions.

Background

Theorem transition determines the critical window for the appearance of outgoing flux in the random recursive tree when the family of car-arrival distributions is bounded. The authors note that the boundedness assumption is used only for the upper bound on the flux and express the belief that the same conclusion should hold for unbounded arrival laws, specifically mentioning Poisson and geometric distributions. Establishing this extension would therefore generalize the paper’s critical-window result beyond bounded car arrivals.