  We consider a stochastic evolutionary model for a phenotype developing
amongst n related species with unknown phylogeny. The unknown tree is modelled
by a Yule process conditioned on n contemporary nodes. The trait value is
assumed to evolve along lineages as an Ornstein-Uhlenbeck process. As a result,
the trait values of the n species form a sample with dependent observations. We
establish three limit theorems for the sample mean corresponding to three
domains for the adaptation rate. In the case of fast adaptation, we show that
for large $n$ the normalized sample mean is approximately normally distributed.
Using these limit theorems, we develop novel confidence interval formulae for
the optimal trait value.
