We consider a stochastic evolutionary model for a phenotype developing amongstn related species with unknown phylogeny. The unknown tree is modelledby a Yule process conditioned on n contemporary nodes. The trait valueis assumed to evolve along lineages as an Ornstein-Uhlenbeck process. As aresult, the trait values of the n species form a sample with dependentobservations. We establish three limit theorems for the sample meancorresponding to three domains for the adaptation rate. In the case of fastadaptation, we show that for large n the normalized sample mean isapproximately normally distributed. Using these limit theorems, we developnovel confidence interval formulae for the optimal trait value.
"Phylogenetic confidence intervals for the optimal trait value." J. Appl. Probab. 52 (4) 1115 - 1132, December 2015. https://doi.org/10.1239/jap/1450802756