  The Pitman-Yor process is a random discrete measure. The random weights or
masses follow the two-parameter Poisson-Dirichlet distribution with parameters
$0<\alpha<1, \theta>-\alpha$. The parameters $\alpha$ and $\theta$ correspond
to the stable and gamma components, respectively. The distribution of atoms is
given by a probability $\nu$. In this article we consider the limit theorems
for the Pitman-Yor process and the two-parameter Poisson-Dirichlet
distribution. These include law of large numbers, fluctuations, and moderate or
large deviation principles. The limiting procedures involve either $\alpha$
tends to zero or one. They arise naturally in genetics and physics such as the
asymptotic coalescence time for explosive branching process and the
approximation to generalized random energy model for disordered system.
