  The Bernstein operator is known as a typical example of positive linear
operators which uniformly approximates continuous functions on $[0, 1]$. In the
present paper, we introduce a multidimensional extension of the Bernstein
operator which is associated with a transition probability of a certain
discrete Markov chain. In particular, we show that the iterate of the
multidimensional Bernstein-type operator uniformly converges to the Feller
semigroup corresponding to the multidimensional Wright-Fisher diffusion process
with mutation arising in the study of population genetics, together with its
rate of convergence. The convergence of process-level is obtained as well.
Moreover, by taking the limit as both the number of iterate and the dimension
of the Bernstein-type operator tend to infinity simultaneously, we prove that
the iterate of the multidimensional Bernstein-type operator uniformly converges
to the Feller semigroup corresponding to a probability measure-valued
Fleming-Viot process with mutation.
