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The Bernstein operator is known as a typical example of positive linearoperators which uniformly approximates continuous functions on $[0, 1]$. In thepresent paper, we introduce a multidimensional extension of the Bernsteinoperator which is associated with a transition probability of a certaindiscrete Markov chain. In particular, we show that the iterate of themultidimensional Bernstein-type operator uniformly converges to the Fellersemigroup corresponding to the multidimensional Wright-Fisher diffusion processwith mutation arising in the study of population genetics, together with itsrate of convergence. The convergence of process-level is obtained as well.Moreover, by taking the limit as both the number of iterate and the dimensionof the Bernstein-type operator tend to infinity simultaneously, we prove thatthe iterate of the multidimensional Bernstein-type operator uniformly convergesto the Feller semigroup corresponding to a probability measure-valuedFleming-Viot process with mutation.

Iterates of multidimensional Bernstein-type operators and diffusion processes in population genetics