  Several results of large deviations are obtained for distributions that are
associated with the Poisson--Dirichlet distribution and the Ewens sampling
formula when the parameter $\theta$ approaches infinity. The motivation for
these results comes from a desire of understanding the exact meaning of
$\theta$ going to infinity. In terms of the law of large numbers and the
central limit theorem, the limiting procedure of $\theta$ going to infinity in
a Poisson--Dirichlet distribution corresponds to a finite allele model where
the mutation rate per individual is fixed and the number of alleles going to
infinity. We call this the finite allele approximation. The first main result
of this article is concerned with the relation between this finite allele
approximation and the Poisson--Dirichlet distribution in terms of large
deviations. Large $\theta$ can also be viewed as a limiting procedure of the
effective population size going to infinity. In the second result a comparison
is done between the sample size and the effective population size based on the
Ewens sampling formula.
