Yule process sample path asymptotics
Résumé
This research report presents two results on sample paths for the Yule process: one fluid limit theorem and one sample path large deviation result. The main interest does not lie in results by themselves but in the understanding the change of measure gives on the way large deviation occurs in the case of non-homogeneous processes. Two different phenomena, with respect to classical large deviations principles, are exhibited. First, the probability decay rate has the same form whatever the speed of divergence from the standard behavior. Second, a large deviation event does not take place by ``twisting'' constantly the transition rates but this deformation is concentrated on an infinitely small proportion of the transition, yet on an infinite number of transitions!