  The Pitman-Yor process is a nonparametric species sampling prior with number
of different species of the order of $n^\sigma$ for some $\sigma>0$. In case of
an atomless true distribution, the asymptotic distribution of the posterior of
the Pitman-Yor process was known but typically inconsistent. In this paper, we
extend this result into a general theorem for arbitrary true distributions. For
discrete distributions the posterior is consistent, but it turns out that there
can be a bias which does not converge to zero at the $\sqrt{n}$ rate. We
propose a bias correction and show that after correcting for the bias, the
posterior distribution will be asymptotically normal. Without the bias
correction, the coverage of the credible sets can be arbitrarily low, and we
illustrate this finding with simulations where we compare the coverage of
corrected and uncorrected credible sets.
