Ancestral reproductive bias in continuous-time branching trees under various sampling schemes

Cheek and Johnston (JMB 86:70, 2023) consider a continuous-time Bienaymé-Galton-Watson tree conditioned on being alive at time T. They study the reproduction events along the ancestral lineage of an individual randomly sampled from all those alive at time T. We give a short proof of an extension of their main results (Cheek and Johnston in JMB 86:70, 2023, Theorems 2.3 and 2.4) to the more general case of Bellman-Harris processes. Our proof also sheds light onto the probabilistic structure of the rate of the reproduction events. A similar method will be applied to explain (i) the different ancestral reproduction bias appearing in work by Geiger (JAP 36:301–309, 1999) and (ii) the fact that the sampling rule considered by Chauvin et al. (SPA 39:117–130, 1991), (Theorem 1) leads to a time homogeneous process along the ancestral lineage.