This paper presents two results on sample paths for the Yule process: one fluid limit theorem and one sample path large deviation result. The main interest is to understand the way large deviation occurs in the case of non-homogeneous processes. There are indeed two new phenomena. First there is no ``typical'' speed of large deviation. Second, the large deviation event is concentrated on a finite interval of time.
Arnaud de La Fortelle. "Yule process sample path asymptotics." Electron. Commun. Probab. 11 193 - 199, 2006. https://doi.org/10.1214/ECP.v11-1215